ftc1lem5 - Metamath Proof Explorer

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Theorem ftc1lem5 25974

Description: Lemma for ftc1 25976. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
ftc1.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1.s	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
ftc1.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
ftc1.i	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
ftc1.c	⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
ftc1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
ftc1.j	⊢ 𝐽 = (𝐿 ↾_t ℝ)
ftc1.k	⊢ 𝐾 = (𝐿 ↾_t 𝐷)
ftc1.l	⊢ 𝐿 = (TopOpen‘ℂ_fld)
ftc1.h	⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))
ftc1.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ftc1.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
ftc1.fc	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))
ftc1.x1	⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))
ftc1.x2	⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅)

**Assertion**
Ref	Expression
ftc1lem5	⊢ ((𝜑 ∧ 𝑋 ≠ 𝐶) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸)

Distinct variable groups: 𝑥,𝑡,𝑦,𝑧,𝐶 𝑡,𝐷,𝑥,𝑦,𝑧 𝑦,𝐺,𝑧 𝑡,𝐴,𝑥,𝑦,𝑧 𝑡,𝐵,𝑥,𝑦,𝑧 𝑡,𝑋,𝑥,𝑧 𝑡,𝐸,𝑦 𝑦,𝐻 𝜑,𝑡,𝑥,𝑦,𝑧 𝑡,𝐹,𝑥,𝑦,𝑧 𝑥,𝐿,𝑦,𝑧 𝑦,𝑅 Allowed substitution hints: 𝑅(𝑥,𝑧,𝑡) 𝐸(𝑥,𝑧) 𝐺(𝑥,𝑡) 𝐻(𝑥,𝑧,𝑡) 𝐽(𝑥,𝑦,𝑧,𝑡) 𝐾(𝑥,𝑦,𝑧,𝑡) 𝐿(𝑡) 𝑋(𝑦)

**Proof of Theorem ftc1lem5**
Step	Hyp	Ref	Expression
1		ftc1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		ftc1.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		iccssre 13438	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
4	1, 2, 3	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5		ftc1.x1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))
6	4, 5	sseldd 3981	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7		ioossicc 13442	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
8		ftc1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
9	7, 8	sselid 3978	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
10	4, 9	sseldd 3981	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
11	6, 10	lttri2d 11383	. . 3 ⊢ (𝜑 → (𝑋 ≠ 𝐶 ↔ (𝑋 < 𝐶 ∨ 𝐶 < 𝑋)))
12	11	biimpa 476	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝐶) → (𝑋 < 𝐶 ∨ 𝐶 < 𝑋))
13	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → 𝑋 ∈ (𝐴[,]𝐵))
14	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → 𝑋 ∈ ℝ)
15		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → 𝑋 < 𝐶)
16	14, 15	ltned 11380	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → 𝑋 ≠ 𝐶)
17		eldifsn 4791	. . . . . . . 8 ⊢ (𝑋 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↔ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑋 ≠ 𝐶))
18	13, 16, 17	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → 𝑋 ∈ ((𝐴[,]𝐵) ∖ {𝐶}))
19		fveq2 6897	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → (𝐺‘𝑧) = (𝐺‘𝑋))
20	19	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → ((𝐺‘𝑧) − (𝐺‘𝐶)) = ((𝐺‘𝑋) − (𝐺‘𝐶)))
21		oveq1 7427	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 − 𝐶) = (𝑋 − 𝐶))
22	20, 21	oveq12d 7438	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = (((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)))
23		ftc1.h	. . . . . . . 8 ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))
24		ovex 7453	. . . . . . . 8 ⊢ (((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)) ∈ V
25	22, 23, 24	fvmpt 7005	. . . . . . 7 ⊢ (𝑋 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) → (𝐻‘𝑋) = (((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)))
26	18, 25	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (𝐻‘𝑋) = (((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)))
27		ftc1.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
28		ftc1.le	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
29		ftc1.s	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
30		ftc1.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
31		ftc1.i	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
32		ftc1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
33		ftc1.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (𝐿 ↾_t ℝ)
34		ftc1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝐿 ↾_t 𝐷)
35		ftc1.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (TopOpen‘ℂ_fld)
36	27, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35	ftc1lem3 25972	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
37	27, 1, 2, 28, 29, 30, 31, 36	ftc1lem2 25970	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
38	37, 5	ffvelcdmd 7095	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℂ)
39	37, 9	ffvelcdmd 7095	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ℂ)
40	38, 39	subcld 11601	. . . . . . . 8 ⊢ (𝜑 → ((𝐺‘𝑋) − (𝐺‘𝐶)) ∈ ℂ)
41	40	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → ((𝐺‘𝑋) − (𝐺‘𝐶)) ∈ ℂ)
42	6	recnd 11272	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ)
43	10	recnd 11272	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
44	42, 43	subcld 11601	. . . . . . . 8 ⊢ (𝜑 → (𝑋 − 𝐶) ∈ ℂ)
45	44	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (𝑋 − 𝐶) ∈ ℂ)
46	42, 43	subeq0ad 11611	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 − 𝐶) = 0 ↔ 𝑋 = 𝐶))
47	46	necon3bid 2982	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 − 𝐶) ≠ 0 ↔ 𝑋 ≠ 𝐶))
48	47	biimpar 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝐶) → (𝑋 − 𝐶) ≠ 0)
49	16, 48	syldan 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (𝑋 − 𝐶) ≠ 0)
50	41, 45, 49	div2negd 12035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (-((𝐺‘𝑋) − (𝐺‘𝐶)) / -(𝑋 − 𝐶)) = (((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)))
51	38, 39	negsubdi2d 11617	. . . . . . . 8 ⊢ (𝜑 → -((𝐺‘𝑋) − (𝐺‘𝐶)) = ((𝐺‘𝐶) − (𝐺‘𝑋)))
52	42, 43	negsubdi2d 11617	. . . . . . . 8 ⊢ (𝜑 → -(𝑋 − 𝐶) = (𝐶 − 𝑋))
53	51, 52	oveq12d 7438	. . . . . . 7 ⊢ (𝜑 → (-((𝐺‘𝑋) − (𝐺‘𝐶)) / -(𝑋 − 𝐶)) = (((𝐺‘𝐶) − (𝐺‘𝑋)) / (𝐶 − 𝑋)))
54	53	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (-((𝐺‘𝑋) − (𝐺‘𝐶)) / -(𝑋 − 𝐶)) = (((𝐺‘𝐶) − (𝐺‘𝑋)) / (𝐶 − 𝑋)))
55	26, 50, 54	3eqtr2d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (𝐻‘𝑋) = (((𝐺‘𝐶) − (𝐺‘𝑋)) / (𝐶 − 𝑋)))
56	55	fvoveq1d 7442	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) = (abs‘((((𝐺‘𝐶) − (𝐺‘𝑋)) / (𝐶 − 𝑋)) − (𝐹‘𝐶))))
57		ftc1.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
58		ftc1.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
59		ftc1.fc	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))
60		ftc1.x2	. . . . 5 ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅)
61	43	subidd 11589	. . . . . . 7 ⊢ (𝜑 → (𝐶 − 𝐶) = 0)
62	61	abs00bd 15270	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐶 − 𝐶)) = 0)
63	58	rpgt0d 13051	. . . . . 6 ⊢ (𝜑 → 0 < 𝑅)
64	62, 63	eqbrtrd 5170	. . . . 5 ⊢ (𝜑 → (abs‘(𝐶 − 𝐶)) < 𝑅)
65	27, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 57, 58, 59, 5, 60, 9, 64	ftc1lem4 25973	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (abs‘((((𝐺‘𝐶) − (𝐺‘𝑋)) / (𝐶 − 𝑋)) − (𝐹‘𝐶))) < 𝐸)
66	56, 65	eqbrtrd 5170	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝐶) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸)
67	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → 𝑋 ∈ (𝐴[,]𝐵))
68	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → 𝐶 ∈ ℝ)
69		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → 𝐶 < 𝑋)
70	68, 69	gtned 11379	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → 𝑋 ≠ 𝐶)
71	67, 70, 17	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → 𝑋 ∈ ((𝐴[,]𝐵) ∖ {𝐶}))
72	71, 25	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → (𝐻‘𝑋) = (((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)))
73	72	fvoveq1d 7442	. . . 4 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) = (abs‘((((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)) − (𝐹‘𝐶))))
74	27, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 57, 58, 59, 9, 64, 5, 60	ftc1lem4 25973	. . . 4 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → (abs‘((((𝐺‘𝑋) − (𝐺‘𝐶)) / (𝑋 − 𝐶)) − (𝐹‘𝐶))) < 𝐸)
75	73, 74	eqbrtrd 5170	. . 3 ⊢ ((𝜑 ∧ 𝐶 < 𝑋) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸)
76	66, 75	jaodan 956	. 2 ⊢ ((𝜑 ∧ (𝑋 < 𝐶 ∨ 𝐶 < 𝑋)) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸)
77	12, 76	syldan 590	1 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝐶) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 ⊆ wss 3947 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 ℂcc 11136 ℝcr 11137 0cc0 11138 < clt 11278 ≤ cle 11279 − cmin 11474 -cneg 11475 / cdiv 11901 ℝ⁺crp 13006 (,)cioo 13356 [,]cicc 13359 abscabs 15213 ↾_t crest 17401 TopOpenctopn 17402 ℂ_fldccnfld 21278 CnP ccnp 23128 𝐿¹cibl 25545 ∫citg 25546

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4243 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-omul 8491 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-cnfld 21279 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cn 23130 df-cnp 23131 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24797 df-ovol 25392 df-vol 25393 df-mbf 25547 df-itg1 25548 df-itg2 25549 df-ibl 25550 df-itg 25551 df-0p 25598

This theorem is referenced by: ftc1lem6 25975

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