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Theorem dvaddbr 25861

Description: The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 25864. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Remove unnecessary hypotheses. (Revised by GG, 10-Apr-2025.)

**Hypotheses**
Ref	Expression
dvadd.f	⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
dvadd.x	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
dvadd.g	⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
dvadd.y	⊢ (𝜑 → 𝑌 ⊆ 𝑆)
dvaddbr.s	⊢ (𝜑 → 𝑆 ⊆ ℂ)
dvadd.bf	⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)
dvadd.bg	⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)
dvadd.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
dvaddbr	⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_f + 𝐺))(𝐾 + 𝐿))

Proof of Theorem dvaddbr Dummy variables 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvadd.bf	. . . . . 6 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)
2		eqid 2728	. . . . . . 7 ⊢ (𝐽 ↾_t 𝑆) = (𝐽 ↾_t 𝑆)
3		dvadd.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4		eqid 2728	. . . . . . 7 ⊢ (𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)))
5		dvaddbr.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
6		dvadd.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
7		dvadd.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
8	2, 3, 4, 5, 6, 7	eldv 25820	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑆 D 𝐹)𝐾 ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
9	1, 8	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶)))
10	9	simpld 494	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑋))
11		dvadd.bg	. . . . . 6 ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)
12		eqid 2728	. . . . . . 7 ⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))
13		dvadd.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)
14		dvadd.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
15	2, 3, 12, 5, 13, 14	eldv 25820	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑆 D 𝐺)𝐿 ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
16	11, 15	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶)))
17	16	simpld 494	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘𝑌))
18	10, 17	elind 4190	. . 3 ⊢ (𝜑 → 𝐶 ∈ (((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∩ ((int‘(𝐽 ↾_t 𝑆))‘𝑌)))
19	3	cnfldtopon 24692	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
20		resttopon 23058	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
21	19, 5, 20	sylancr 586	. . . . 5 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
22		topontop 22808	. . . . 5 ⊢ ((𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆) → (𝐽 ↾_t 𝑆) ∈ Top)
23	21, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ Top)
24		toponuni 22809	. . . . . 6 ⊢ ((𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ (𝐽 ↾_t 𝑆))
25	21, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 = ∪ (𝐽 ↾_t 𝑆))
26	7, 25	sseqtrd 4018	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ∪ (𝐽 ↾_t 𝑆))
27	14, 25	sseqtrd 4018	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ (𝐽 ↾_t 𝑆))
28		eqid 2728	. . . . 5 ⊢ ∪ (𝐽 ↾_t 𝑆) = ∪ (𝐽 ↾_t 𝑆)
29	28	ntrin 22958	. . . 4 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ (𝐽 ↾_t 𝑆) ∧ 𝑌 ⊆ ∪ (𝐽 ↾_t 𝑆)) → ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∩ ((int‘(𝐽 ↾_t 𝑆))‘𝑌)))
30	23, 26, 27, 29	syl3anc 1369	. . 3 ⊢ (𝜑 → ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾_t 𝑆))‘𝑋) ∩ ((int‘(𝐽 ↾_t 𝑆))‘𝑌)))
31	18, 30	eleqtrrd 2832	. 2 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)))
32		inss1 4224	. . . . . . 7 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋
33		ssdif 4135	. . . . . . 7 ⊢ ((𝑋 ∩ 𝑌) ⊆ 𝑋 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑋 ∖ {𝐶}))
34	32, 33	mp1i 13	. . . . . 6 ⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑋 ∖ {𝐶}))
35	34	sselda 3978	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ (𝑋 ∖ {𝐶}))
36	7, 5	sstrd 3988	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
37	28	ntrss2 22954	. . . . . . . 8 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ (𝐽 ↾_t 𝑆)) → ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ⊆ 𝑋)
38	23, 26, 37	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((int‘(𝐽 ↾_t 𝑆))‘𝑋) ⊆ 𝑋)
39	38, 10	sseldd 3979	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
40	6, 36, 39	dvlem 25818	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
41	35, 40	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
42		inss2 4225	. . . . . . 7 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌
43		ssdif 4135	. . . . . . 7 ⊢ ((𝑋 ∩ 𝑌) ⊆ 𝑌 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑌 ∖ {𝐶}))
44	42, 43	mp1i 13	. . . . . 6 ⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑌 ∖ {𝐶}))
45	44	sselda 3978	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ (𝑌 ∖ {𝐶}))
46	14, 5	sstrd 3988	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
47	28	ntrss2 22954	. . . . . . . 8 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ 𝑌 ⊆ ∪ (𝐽 ↾_t 𝑆)) → ((int‘(𝐽 ↾_t 𝑆))‘𝑌) ⊆ 𝑌)
48	23, 27, 47	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((int‘(𝐽 ↾_t 𝑆))‘𝑌) ⊆ 𝑌)
49	48, 17	sseldd 3979	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
50	13, 46, 49	dvlem 25818	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
51	45, 50	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
52		ssidd 4001	. . . 4 ⊢ (𝜑 → ℂ ⊆ ℂ)
53		txtopon 23488	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐽 ∈ (TopOn‘ℂ)) → (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ)))
54	19, 19, 53	mp2an 691	. . . . 5 ⊢ (𝐽 ×_t 𝐽) ∈ (TopOn‘(ℂ × ℂ))
55	54	toponrestid 22816	. . . 4 ⊢ (𝐽 ×_t 𝐽) = ((𝐽 ×_t 𝐽) ↾_t (ℂ × ℂ))
56	9	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
57	40	fmpttd 7119	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))):(𝑋 ∖ {𝐶})⟶ℂ)
58	36	ssdifssd 4138	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ {𝐶}) ⊆ ℂ)
59		eqid 2728	. . . . . . 7 ⊢ (𝐽 ↾_t ((𝑋 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾_t ((𝑋 ∖ {𝐶}) ∪ {𝐶}))
60	32, 7	sstrid 3989	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ 𝑆)
61	60, 25	sseqtrd 4018	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ ∪ (𝐽 ↾_t 𝑆))
62		difssd 4128	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋) ⊆ ∪ (𝐽 ↾_t 𝑆))
63	61, 62	unssd 4182	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋)) ⊆ ∪ (𝐽 ↾_t 𝑆))
64		ssun1 4168	. . . . . . . . . . . . 13 ⊢ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))
65	64	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋)))
66	28	ntrss 22952	. . . . . . . . . . . 12 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋)) ⊆ ∪ (𝐽 ↾_t 𝑆) ∧ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))) → ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))))
67	23, 63, 65, 66	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))))
68	67, 31	sseldd 3979	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))))
69	68, 39	elind 4190	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))) ∩ 𝑋))
70	32	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ 𝑋)
71		eqid 2728	. . . . . . . . . . . 12 ⊢ ((𝐽 ↾_t 𝑆) ↾_t 𝑋) = ((𝐽 ↾_t 𝑆) ↾_t 𝑋)
72	28, 71	restntr 23079	. . . . . . . . . . 11 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ (𝐽 ↾_t 𝑆) ∧ (𝑋 ∩ 𝑌) ⊆ 𝑋) → ((int‘((𝐽 ↾_t 𝑆) ↾_t 𝑋))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))) ∩ 𝑋))
73	23, 26, 70, 72	syl3anc 1369	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘((𝐽 ↾_t 𝑆) ↾_t 𝑋))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))) ∩ 𝑋))
74	3	cnfldtop 24693	. . . . . . . . . . . . . 14 ⊢ 𝐽 ∈ Top
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
76		cnex 11213	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
77		ssexg 5317	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
78	5, 76, 77	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ V)
79		restabs 23062	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ V) → ((𝐽 ↾_t 𝑆) ↾_t 𝑋) = (𝐽 ↾_t 𝑋))
80	75, 7, 78, 79	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽 ↾_t 𝑆) ↾_t 𝑋) = (𝐽 ↾_t 𝑋))
81	80	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝜑 → (int‘((𝐽 ↾_t 𝑆) ↾_t 𝑋)) = (int‘(𝐽 ↾_t 𝑋)))
82	81	fveq1d 6893	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘((𝐽 ↾_t 𝑆) ↾_t 𝑋))‘(𝑋 ∩ 𝑌)) = ((int‘(𝐽 ↾_t 𝑋))‘(𝑋 ∩ 𝑌)))
83	73, 82	eqtr3d 2770	. . . . . . . . 9 ⊢ (𝜑 → (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑋))) ∩ 𝑋) = ((int‘(𝐽 ↾_t 𝑋))‘(𝑋 ∩ 𝑌)))
84	69, 83	eleqtrd 2831	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑋))‘(𝑋 ∩ 𝑌)))
85		undif1 4471	. . . . . . . . . . . 12 ⊢ ((𝑋 ∖ {𝐶}) ∪ {𝐶}) = (𝑋 ∪ {𝐶})
86	39	snssd 4808	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐶} ⊆ 𝑋)
87		ssequn2 4179	. . . . . . . . . . . . 13 ⊢ ({𝐶} ⊆ 𝑋 ↔ (𝑋 ∪ {𝐶}) = 𝑋)
88	86, 87	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∪ {𝐶}) = 𝑋)
89	85, 88	eqtrid 2780	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ∖ {𝐶}) ∪ {𝐶}) = 𝑋)
90	89	oveq2d 7430	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝑋 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾_t 𝑋))
91	90	fveq2d 6895	. . . . . . . . 9 ⊢ (𝜑 → (int‘(𝐽 ↾_t ((𝑋 ∖ {𝐶}) ∪ {𝐶}))) = (int‘(𝐽 ↾_t 𝑋)))
92		undif1 4471	. . . . . . . . . 10 ⊢ (((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶}) = ((𝑋 ∩ 𝑌) ∪ {𝐶})
93	39, 49	elind 4190	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ (𝑋 ∩ 𝑌))
94	93	snssd 4808	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐶} ⊆ (𝑋 ∩ 𝑌))
95		ssequn2 4179	. . . . . . . . . . 11 ⊢ ({𝐶} ⊆ (𝑋 ∩ 𝑌) ↔ ((𝑋 ∩ 𝑌) ∪ {𝐶}) = (𝑋 ∩ 𝑌))
96	94, 95	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∪ {𝐶}) = (𝑋 ∩ 𝑌))
97	92, 96	eqtrid 2780	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶}) = (𝑋 ∩ 𝑌))
98	91, 97	fveq12d 6898	. . . . . . . 8 ⊢ (𝜑 → ((int‘(𝐽 ↾_t ((𝑋 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶})) = ((int‘(𝐽 ↾_t 𝑋))‘(𝑋 ∩ 𝑌)))
99	84, 98	eleqtrrd 2832	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t ((𝑋 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶})))
100	57, 34, 58, 3, 59, 99	limcres 25808	. . . . . 6 ⊢ (𝜑 → (((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
101	34	resmptd 6038	. . . . . . 7 ⊢ (𝜑 → ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))))
102	101	oveq1d 7429	. . . . . 6 ⊢ (𝜑 → (((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
103	100, 102	eqtr3d 2770	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
104	56, 103	eleqtrd 2831	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
105	16	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
106	50	fmpttd 7119	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))):(𝑌 ∖ {𝐶})⟶ℂ)
107	46	ssdifssd 4138	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∖ {𝐶}) ⊆ ℂ)
108		eqid 2728	. . . . . . 7 ⊢ (𝐽 ↾_t ((𝑌 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾_t ((𝑌 ∖ {𝐶}) ∪ {𝐶}))
109		difssd 4128	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌) ⊆ ∪ (𝐽 ↾_t 𝑆))
110	61, 109	unssd 4182	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌)) ⊆ ∪ (𝐽 ↾_t 𝑆))
111		ssun1 4168	. . . . . . . . . . . . 13 ⊢ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))
112	111	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌)))
113	28	ntrss 22952	. . . . . . . . . . . 12 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌)) ⊆ ∪ (𝐽 ↾_t 𝑆) ∧ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))) → ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))))
114	23, 110, 112, 113	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))))
115	114, 31	sseldd 3979	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))))
116	115, 49	elind 4190	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))) ∩ 𝑌))
117	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ 𝑌)
118		eqid 2728	. . . . . . . . . . . 12 ⊢ ((𝐽 ↾_t 𝑆) ↾_t 𝑌) = ((𝐽 ↾_t 𝑆) ↾_t 𝑌)
119	28, 118	restntr 23079	. . . . . . . . . . 11 ⊢ (((𝐽 ↾_t 𝑆) ∈ Top ∧ 𝑌 ⊆ ∪ (𝐽 ↾_t 𝑆) ∧ (𝑋 ∩ 𝑌) ⊆ 𝑌) → ((int‘((𝐽 ↾_t 𝑆) ↾_t 𝑌))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))) ∩ 𝑌))
120	23, 27, 117, 119	syl3anc 1369	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘((𝐽 ↾_t 𝑆) ↾_t 𝑌))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))) ∩ 𝑌))
121		restabs 23062	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑆 ∧ 𝑆 ∈ V) → ((𝐽 ↾_t 𝑆) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
122	75, 14, 78, 121	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽 ↾_t 𝑆) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
123	122	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝜑 → (int‘((𝐽 ↾_t 𝑆) ↾_t 𝑌)) = (int‘(𝐽 ↾_t 𝑌)))
124	123	fveq1d 6893	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘((𝐽 ↾_t 𝑆) ↾_t 𝑌))‘(𝑋 ∩ 𝑌)) = ((int‘(𝐽 ↾_t 𝑌))‘(𝑋 ∩ 𝑌)))
125	120, 124	eqtr3d 2770	. . . . . . . . 9 ⊢ (𝜑 → (((int‘(𝐽 ↾_t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾_t 𝑆) ∖ 𝑌))) ∩ 𝑌) = ((int‘(𝐽 ↾_t 𝑌))‘(𝑋 ∩ 𝑌)))
126	116, 125	eleqtrd 2831	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t 𝑌))‘(𝑋 ∩ 𝑌)))
127		undif1 4471	. . . . . . . . . . . 12 ⊢ ((𝑌 ∖ {𝐶}) ∪ {𝐶}) = (𝑌 ∪ {𝐶})
128	49	snssd 4808	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐶} ⊆ 𝑌)
129		ssequn2 4179	. . . . . . . . . . . . 13 ⊢ ({𝐶} ⊆ 𝑌 ↔ (𝑌 ∪ {𝐶}) = 𝑌)
130	128, 129	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∪ {𝐶}) = 𝑌)
131	127, 130	eqtrid 2780	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌 ∖ {𝐶}) ∪ {𝐶}) = 𝑌)
132	131	oveq2d 7430	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝑌 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾_t 𝑌))
133	132	fveq2d 6895	. . . . . . . . 9 ⊢ (𝜑 → (int‘(𝐽 ↾_t ((𝑌 ∖ {𝐶}) ∪ {𝐶}))) = (int‘(𝐽 ↾_t 𝑌)))
134	133, 97	fveq12d 6898	. . . . . . . 8 ⊢ (𝜑 → ((int‘(𝐽 ↾_t ((𝑌 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶})) = ((int‘(𝐽 ↾_t 𝑌))‘(𝑋 ∩ 𝑌)))
135	126, 134	eleqtrrd 2832	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾_t ((𝑌 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶})))
136	106, 44, 107, 3, 108, 135	limcres 25808	. . . . . 6 ⊢ (𝜑 → (((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
137	44	resmptd 6038	. . . . . . 7 ⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
138	137	oveq1d 7429	. . . . . 6 ⊢ (𝜑 → (((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) lim_ℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
139	136, 138	eqtr3d 2770	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
140	105, 139	eleqtrd 2831	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
141	3	addcn 24774	. . . . 5 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
142	5, 6, 7	dvcl 25821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ)
143	1, 142	mpdan 686	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
144	5, 13, 14	dvcl 25821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶(𝑆 D 𝐺)𝐿) → 𝐿 ∈ ℂ)
145	11, 144	mpdan 686	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
146	143, 145	opelxpd 5711	. . . . 5 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ))
147	54	toponunii 22811	. . . . . 6 ⊢ (ℂ × ℂ) = ∪ (𝐽 ×_t 𝐽)
148	147	cncnpi 23175	. . . . 5 ⊢ (( + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽) ∧ ⟨𝐾, 𝐿⟩ ∈ (ℂ × ℂ)) → + ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
149	141, 146, 148	sylancr 586	. . . 4 ⊢ (𝜑 → + ∈ (((𝐽 ×_t 𝐽) CnP 𝐽)‘⟨𝐾, 𝐿⟩))
150	41, 51, 52, 52, 3, 55, 104, 140, 149	limccnp2 25814	. . 3 ⊢ (𝜑 → (𝐾 + 𝐿) ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
151		eldifi 4122	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ∈ (𝑋 ∩ 𝑌))
152	151	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ (𝑋 ∩ 𝑌))
153	6	ffnd 6717	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn 𝑋)
154	153	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝐹 Fn 𝑋)
155	13	ffnd 6717	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 Fn 𝑌)
156	155	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝐺 Fn 𝑌)
157		ssexg 5317	. . . . . . . . . . . . 13 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
158	36, 76, 157	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ V)
159	158	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑋 ∈ V)
160		ssexg 5317	. . . . . . . . . . . . 13 ⊢ ((𝑌 ⊆ ℂ ∧ ℂ ∈ V) → 𝑌 ∈ V)
161	46, 76, 160	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ V)
162	161	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑌 ∈ V)
163		eqid 2728	. . . . . . . . . . 11 ⊢ (𝑋 ∩ 𝑌) = (𝑋 ∩ 𝑌)
164		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = (𝐹‘𝑧))
165		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) = (𝐺‘𝑧))
166	154, 156, 159, 162, 163, 164, 165	ofval 7690	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝑧 ∈ (𝑋 ∩ 𝑌)) → ((𝐹 ∘_f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
167	152, 166	mpdan 686	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐹 ∘_f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
168		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝐶 ∈ 𝑋) → (𝐹‘𝐶) = (𝐹‘𝐶))
169		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝐶 ∈ 𝑌) → (𝐺‘𝐶) = (𝐺‘𝐶))
170	154, 156, 159, 162, 163, 168, 169	ofval 7690	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝐶 ∈ (𝑋 ∩ 𝑌)) → ((𝐹 ∘_f + 𝐺)‘𝐶) = ((𝐹‘𝐶) + (𝐺‘𝐶)))
171	93, 170	mpidan 688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐹 ∘_f + 𝐺)‘𝐶) = ((𝐹‘𝐶) + (𝐺‘𝐶)))
172	167, 171	oveq12d 7432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) = (((𝐹‘𝑧) + (𝐺‘𝑧)) − ((𝐹‘𝐶) + (𝐺‘𝐶))))
173		difss 4127	. . . . . . . . . . . 12 ⊢ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑋 ∩ 𝑌)
174	173, 32	sstri 3987	. . . . . . . . . . 11 ⊢ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ 𝑋
175	174	sseli 3974	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ∈ 𝑋)
176		ffvelcdm 7085	. . . . . . . . . 10 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
177	6, 175, 176	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐹‘𝑧) ∈ ℂ)
178	173, 42	sstri 3987	. . . . . . . . . . 11 ⊢ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ 𝑌
179	178	sseli 3974	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ∈ 𝑌)
180		ffvelcdm 7085	. . . . . . . . . 10 ⊢ ((𝐺:𝑌⟶ℂ ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ ℂ)
181	13, 179, 180	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ)
182	6, 39	ffvelcdmd 7089	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
183	182	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ)
184	13, 49	ffvelcdmd 7089	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ℂ)
185	184	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ)
186	177, 181, 183, 185	addsub4d 11642	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹‘𝑧) + (𝐺‘𝑧)) − ((𝐹‘𝐶) + (𝐺‘𝐶))) = (((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶))))
187	172, 186	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) = (((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶))))
188	187	oveq1d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = ((((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶))) / (𝑧 − 𝐶)))
189	177, 183	subcld 11595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐹‘𝑧) − (𝐹‘𝐶)) ∈ ℂ)
190	181, 185	subcld 11595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ)
191	174, 36	sstrid 3989	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ ℂ)
192	191	sselda 3978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ ℂ)
193	36, 39	sseldd 3979	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
194	193	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝐶 ∈ ℂ)
195	192, 194	subcld 11595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ)
196		eldifsni 4789	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ≠ 𝐶)
197	196	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ≠ 𝐶)
198	192, 194, 197	subne0d 11604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0)
199	189, 190, 195, 198	divdird 12052	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶))) / (𝑧 − 𝐶)) = ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
200	188, 199	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))
201	200	mpteq2dva 5242	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))))
202	201	oveq1d 7429	. . 3 ⊢ (𝜑 → ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) lim_ℂ 𝐶))
203	150, 202	eleqtrrd 2832	. 2 ⊢ (𝜑 → (𝐾 + 𝐿) ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
204		eqid 2728	. . 3 ⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶)))
205		addcl 11214	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
206	205	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
207	206, 6, 13, 158, 161, 163	off 7697	. . 3 ⊢ (𝜑 → (𝐹 ∘_f + 𝐺):(𝑋 ∩ 𝑌)⟶ℂ)
208	2, 3, 204, 5, 207, 60	eldv 25820	. 2 ⊢ (𝜑 → (𝐶(𝑆 D (𝐹 ∘_f + 𝐺))(𝐾 + 𝐿) ↔ (𝐶 ∈ ((int‘(𝐽 ↾_t 𝑆))‘(𝑋 ∩ 𝑌)) ∧ (𝐾 + 𝐿) ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘_f + 𝐺)‘𝑧) − ((𝐹 ∘_f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))))
209	31, 203, 208	mpbir2and 712	1 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_f + 𝐺))(𝐾 + 𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ∖ cdif 3942 ∪ cun 3943 ∩ cin 3944 ⊆ wss 3945 {csn 4624 ⟨cop 4630 ∪ cuni 4903 class class class wbr 5142 ↦ cmpt 5225 × cxp 5670 ↾ cres 5674 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∘_f cof 7677 ℂcc 11130 + caddc 11135 − cmin 11468 / cdiv 11895 ↾_t crest 17395 TopOpenctopn 17396 ℂ_fldccnfld 21272 Topctop 22788 TopOnctopon 22805 intcnt 22914 Cn ccn 23121 CnP ccnp 23122 ×_t ctx 23457 lim_ℂ climc 25784 D cdv 25785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-cn 23124 df-cnp 23125 df-tx 23459 df-hmeo 23652 df-xms 24219 df-ms 24220 df-tms 24221 df-limc 25788 df-dv 25789

This theorem is referenced by: dvadd 25864 dvaddf 25866

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