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Theorem rrxcph 25313

Description: Generalized Euclidean real spaces are subcomplex pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)

**Hypotheses**
Ref	Expression
rrxval.r	⊢ 𝐻 = (ℝ^‘𝐼)
rrxbase.b	⊢ 𝐵 = (Base‘𝐻)

**Assertion**
Ref	Expression
rrxcph	⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil)

Proof of Theorem rrxcph Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxval.r	. . 3 ⊢ 𝐻 = (ℝ^‘𝐼)
2	1	rrxval 25308	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2728	. . 3 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2728	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2728	. . 3 ⊢ (Scalar‘(ℝ_fld freeLMod 𝐼)) = (Scalar‘(ℝ_fld freeLMod 𝐼))
6		eqid 2728	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
7		rebase 21531	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
8		remulr 21536	. . . 4 ⊢ · = (._r‘ℝ_fld)
9		eqid 2728	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
10		eqid 2728	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
11		re0g 21537	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
12		refldcj 21545	. . . 4 ⊢ ∗ = (*_𝑟‘ℝ_fld)
13		refld 21544	. . . . 5 ⊢ ℝ_fld ∈ Field
14	13	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld ∈ Field)
15		fconstmpt 5734	. . . . 5 ⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)
16	6, 7, 4	frlmbasf 21687	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓:𝐼⟶ℝ)
17	16	ffnd 6717	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 Fn 𝐼)
18	17	3adant3 1130	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → 𝑓 Fn 𝐼)
19		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝐼 ∈ 𝑉)
20	13	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ℝ_fld ∈ Field)
21		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
22	6, 7, 8, 4, 9	frlmipval 21706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ∈ 𝑉 ∧ ℝ_fld ∈ Field) ∧ (𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)))
23	19, 20, 21, 21, 22	syl22anc 838	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)))
24		inidm 4214	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∩ 𝐼) = 𝐼
25		eqidd 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
26	17, 17, 19, 19, 24, 25, 25	offval 7688	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑓‘𝑥))))
27	16	ffvelcdmda 7088	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
28	27, 27	remulcld 11268	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
29	26, 28	fmpt3d 7120	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓):𝐼⟶ℝ)
30		ovexd 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) ∈ V)
31	29	ffund 6720	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → Fun (𝑓 ∘_f · 𝑓))
32	6, 11, 4	frlmbasfsupp 21685	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 finSupp 0)
33		0red 11241	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ∈ ℝ)
34		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
35	34	recnd 11266	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
36	35	mul02d 11436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → (0 · 𝑥) = 0)
37	19, 33, 16, 16, 36	suppofss1d 8203	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ (𝑓 supp 0))
38		fsuppsssupp 9398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓 ∘_f · 𝑓) ∈ V ∧ Fun (𝑓 ∘_f · 𝑓)) ∧ (𝑓 finSupp 0 ∧ ((𝑓 ∘_f · 𝑓) supp 0) ⊆ (𝑓 supp 0))) → (𝑓 ∘_f · 𝑓) finSupp 0)
39	30, 31, 32, 37, 38	syl22anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) finSupp 0)
40		regsumsupp 21547	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∘_f · 𝑓):𝐼⟶ℝ ∧ (𝑓 ∘_f · 𝑓) finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)) = Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓 ∘_f · 𝑓)‘𝑥))
41	29, 39, 19, 40	syl3anc 1369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)) = Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓 ∘_f · 𝑓)‘𝑥))
42		suppssdm 8175	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
43	42, 16	fssdm 6736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ⊆ 𝐼)
44	37, 43	sstrd 3988	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼)
45	44	sselda 3978	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → 𝑥 ∈ 𝐼)
46	17, 17, 19, 19, 24, 25, 25	ofval 7690	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓 ∘_f · 𝑓)‘𝑥) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
47	45, 46	syldan 590	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓 ∘_f · 𝑓)‘𝑥) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
48	47	sumeq2dv 15675	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓 ∘_f · 𝑓)‘𝑥) = Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
49	41, 48	eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)) = Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
50	23, 49	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
51	50	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
52		simp3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0)
53	51, 52	eqtr3d 2770	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)) = 0)
54	32	fsuppimpd 9387	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ∈ Fin)
55	54, 37	ssfid 9285	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ∈ Fin)
56	45, 28	syldan 590	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
57	27	msqge0d 11806	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((𝑓‘𝑥) · (𝑓‘𝑥)))
58	45, 57	syldan 590	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → 0 ≤ ((𝑓‘𝑥) · (𝑓‘𝑥)))
59	55, 56, 58	fsum00 15770	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)) = 0 ↔ ∀𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)) = 0))
60	59	3adant3 1130	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → (Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)) = 0 ↔ ∀𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)) = 0))
61	53, 60	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → ∀𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)) = 0)
62	61	r19.21bi 3244	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓‘𝑥) · (𝑓‘𝑥)) = 0)
63	62	adantlr 714	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓‘𝑥) · (𝑓‘𝑥)) = 0)
64	27	3adantl3 1166	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
65	64	recnd 11266	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
66	65, 65	mul0ord 11888	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) · (𝑓‘𝑥)) = 0 ↔ ((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0)))
67	66	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → (((𝑓‘𝑥) · (𝑓‘𝑥)) = 0 ↔ ((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0)))
68	63, 67	mpbid 231	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0))
69		oridm 903	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0) ↔ (𝑓‘𝑥) = 0)
70	68, 69	sylib 217	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → (𝑓‘𝑥) = 0)
71	29	3adant3 1130	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → (𝑓 ∘_f · 𝑓):𝐼⟶ℝ)
72	71	adantr 480	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (𝑓 ∘_f · 𝑓):𝐼⟶ℝ)
73		ssidd 4001	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ ((𝑓 ∘_f · 𝑓) supp 0))
74		simpl1 1189	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑉)
75		0red 11241	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → 0 ∈ ℝ)
76	72, 73, 74, 75	suppssr 8194	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) → ((𝑓 ∘_f · 𝑓)‘𝑥) = 0)
77	46	3adantl3 1166	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → ((𝑓 ∘_f · 𝑓)‘𝑥) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
78	77	eqeq1d 2730	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (((𝑓 ∘_f · 𝑓)‘𝑥) = 0 ↔ ((𝑓‘𝑥) · (𝑓‘𝑥)) = 0))
79	78, 66	bitrd 279	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (((𝑓 ∘_f · 𝑓)‘𝑥) = 0 ↔ ((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0)))
80	79, 69	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (((𝑓 ∘_f · 𝑓)‘𝑥) = 0 ↔ (𝑓‘𝑥) = 0))
81	80	biimpa 476	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ ((𝑓 ∘_f · 𝑓)‘𝑥) = 0) → (𝑓‘𝑥) = 0)
82	76, 81	syldan 590	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) → (𝑓‘𝑥) = 0)
83		undif 4477	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼 ↔ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) = 𝐼)
84	44, 83	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) = 𝐼)
85	84	eleq2d 2815	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑥 ∈ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) ↔ 𝑥 ∈ 𝐼))
86	85	3adant3 1130	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → (𝑥 ∈ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) ↔ 𝑥 ∈ 𝐼))
87	86	biimpar 477	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))))
88		elun 4144	. . . . . . . . 9 ⊢ (𝑥 ∈ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) ↔ (𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0) ∨ 𝑥 ∈ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))))
89	87, 88	sylib 217	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0) ∨ 𝑥 ∈ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))))
90	70, 82, 89	mpjaodan 957	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = 0)
91	90	ralrimiva 3142	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0)
92		fconstfv 7218	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0))
93		c0ex 11232	. . . . . . . 8 ⊢ 0 ∈ V
94	93	fconst2 7211	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ 𝑓 = (𝐼 × {0}))
95	92, 94	sylbb1 236	. . . . . 6 ⊢ ((𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0) → 𝑓 = (𝐼 × {0}))
96	18, 91, 95	syl2anc 583	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → 𝑓 = (𝐼 × {0}))
97		isfld 20628	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
98	13, 97	mpbi 229	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing)
99	98	simpli 483	. . . . . . . . 9 ⊢ ℝ_fld ∈ DivRing
100		drngring 20624	. . . . . . . . 9 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
101	99, 100	ax-mp 5	. . . . . . . 8 ⊢ ℝ_fld ∈ Ring
102	6, 11	frlm0 21681	. . . . . . . 8 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
103	101, 102	mpan 689	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
104	103, 15	eqtr3di 2783	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = (𝑥 ∈ 𝐼 ↦ 0))
105	104	3ad2ant1 1131	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → (0_g‘(ℝ_fld freeLMod 𝐼)) = (𝑥 ∈ 𝐼 ↦ 0))
106	15, 96, 105	3eqtr4a 2794	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → 𝑓 = (0_g‘(ℝ_fld freeLMod 𝐼)))
107		cjre 15112	. . . . 5 ⊢ (𝑥 ∈ ℝ → (∗‘𝑥) = 𝑥)
108	107	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ ℝ) → (∗‘𝑥) = 𝑥)
109		id 22	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
110	6, 7, 8, 4, 9, 10, 11, 12, 14, 106, 108, 109	frlmphl 21708	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ PreHil)
111	6	frlmsca 21680	. . . . 5 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
112	13, 111	mpan 689	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
113		df-refld 21530	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
114	112, 113	eqtr3di 2783	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(ℝ_fld freeLMod 𝐼)) = (ℂ_fld ↾_s ℝ))
115		simpr1 1192	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → 𝑓 ∈ ℝ)
116		simpr3 1194	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → 0 ≤ 𝑓)
117	115, 116	resqrtcld 15390	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → (√‘𝑓) ∈ ℝ)
118	55, 56, 58	fsumge0 15767	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
119	118, 49	breqtrrd 5170	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)))
120	119, 23	breqtrrd 5170	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))
121	3, 4, 5, 110, 114, 9, 117, 120	tcphcph 25158	. 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ ℂPreHil)
122	2, 121	eqeltrd 2829	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∖ cdif 3942 ∪ cun 3943 ⊆ wss 3945 {csn 4624 class class class wbr 5142 ↦ cmpt 5225 × cxp 5670 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∘_f cof 7677 supp csupp 8159 finSupp cfsupp 9379 ℝcr 11131 0cc0 11132 · cmul 11137 ≤ cle 11273 ∗ccj 15069 Σcsu 15658 Basecbs 17173 ↾_s cress 17202 Scalarcsca 17229 ·_𝑖cip 17231 0_gc0g 17414 Σ_g cgsu 17415 Ringcrg 20166 CRingccrg 20167 DivRingcdr 20617 Fieldcfield 20618 ℂ_fldccnfld 21272 ℝ_fldcrefld 21529 freeLMod cfrlm 21673 ℂPreHilccph 25087 toℂPreHilctcph 25088 ℝ^crrx 25304

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-inf2 9658 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 ax-mulf 11212

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-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-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 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-ico 13356 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-clim 15458 df-sum 15659 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-prds 17422 df-pws 17424 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-rhm 20404 df-subrng 20476 df-subrg 20501 df-drng 20619 df-field 20620 df-abv 20690 df-staf 20718 df-srng 20719 df-lmod 20738 df-lss 20809 df-lmhm 20900 df-lvec 20981 df-sra 21051 df-rgmod 21052 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-cnfld 21273 df-refld 21530 df-phl 21551 df-dsmm 21659 df-frlm 21674 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-xms 24219 df-ms 24220 df-nm 24484 df-ngp 24485 df-tng 24486 df-nrg 24487 df-nlm 24488 df-clm 24983 df-cph 25089 df-tcph 25090 df-rrx 25306

This theorem is referenced by: rrxngp 45667

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