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Theorem rrxcph 24461

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 24456	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2738	. . 3 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2738	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2738	. . 3 ⊢ (Scalar‘(ℝ_fld freeLMod 𝐼)) = (Scalar‘(ℝ_fld freeLMod 𝐼))
6		eqid 2738	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
7		rebase 20723	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
8		remulr 20728	. . . 4 ⊢ · = (._r‘ℝ_fld)
9		eqid 2738	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
10		eqid 2738	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
11		re0g 20729	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
12		refldcj 20737	. . . 4 ⊢ ∗ = (*_𝑟‘ℝ_fld)
13		refld 20736	. . . . 5 ⊢ ℝ_fld ∈ Field
14	13	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld ∈ Field)
15		fconstmpt 5640	. . . . 5 ⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)
16	6, 7, 4	frlmbasf 20877	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓:𝐼⟶ℝ)
17	16	ffnd 6585	. . . . . . 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 20896	. . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∩ 𝐼) = 𝐼
25		eqidd 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
26	17, 17, 19, 19, 24, 25, 25	offval 7520	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑓‘𝑥))))
27	16	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
28	27, 27	remulcld 10936	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
29	26, 28	fmpt3d 6972	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓):𝐼⟶ℝ)
30		ovexd 7290	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) ∈ V)
31	29	ffund 6588	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → Fun (𝑓 ∘_f · 𝑓))
32	6, 11, 4	frlmbasfsupp 20875	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 finSupp 0)
33		0red 10909	. . . . . . . . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
36	35	mul02d 11103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → (0 · 𝑥) = 0)
37	19, 33, 16, 16, 36	suppofss1d 7991	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ (𝑓 supp 0))
38		fsuppsssupp 9074	. . . . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) finSupp 0)
40		regsumsupp 20739	. . . . . . . . . . . . . . . . . 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 7964	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
43	42, 16	fssdm 6604	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ⊆ 𝐼)
44	37, 43	sstrd 3927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼)
45	44	sselda 3917	. . . . . . . . . . . . . . . . . . 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 7522	. . . . . . . . . . . . . . . . . . 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 15343	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 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 9065	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ∈ Fin)
55	54, 37	ssfid 8971	. . . . . . . . . . . . . . 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 11473	. . . . . . . . . . . . . . . 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 15438	. . . . . . . . . . . . . 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 3132	. . . . . . . . . . 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 711	. . . . . . . . . 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 10934	. . . . . . . . . . . 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 11555	. . . . . . . . . . 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 901	. . . . . . . . 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 3940	. . . . . . . . . 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 10909	. . . . . . . . . 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 7983	. . . . . . . . 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 2740	. . . . . . . . . . . 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 278	. . . . . . . . . . 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 286	. . . . . . . . . 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 4412	. . . . . . . . . . . . 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 2824	. . . . . . . . . . 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 4079	. . . . . . . . 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 955	. . . . . . 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 3107	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0)
92		fconstfv 7070	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0))
93		c0ex 10900	. . . . . . . 8 ⊢ 0 ∈ V
94	93	fconst2 7062	. . . . . . 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 19915	. . . . . . . . . . 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 19913	. . . . . . . . 9 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
101	99, 100	ax-mp 5	. . . . . . . 8 ⊢ ℝ_fld ∈ Ring
102	6, 11	frlm0 20871	. . . . . . . 8 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
103	101, 102	mpan 686	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
104	103, 15	eqtr3di 2794	. . . . . 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 2805	. . . 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 14778	. . . . 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 20898	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ PreHil)
111	6	frlmsca 20870	. . . . 5 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
112	13, 111	mpan 686	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
113		df-refld 20722	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
114	112, 113	eqtr3di 2794	. . 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 15057	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → (√‘𝑓) ∈ ℝ)
118	55, 56, 58	fsumge0 15435	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
119	118, 49	breqtrrd 5098	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)))
120	119, 23	breqtrrd 5098	. . 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 24306	. 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ ℂPreHil)
122	2, 121	eqeltrd 2839	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 supp csupp 7948 finSupp cfsupp 9058 ℝcr 10801 0cc0 10802 · cmul 10807 ≤ cle 10941 ∗ccj 14735 Σcsu 15325 Basecbs 16840 ↾_s cress 16867 Scalarcsca 16891 ·_𝑖cip 16893 0_gc0g 17067 Σ_g cgsu 17068 Ringcrg 19698 CRingccrg 19699 DivRingcdr 19906 Fieldcfield 19907 ℂ_fldccnfld 20510 ℝ_fldcrefld 20721 freeLMod cfrlm 20863 ℂPreHilccph 24235 toℂPreHilctcph 24236 ℝ^crrx 24452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-prds 17075 df-pws 17077 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-field 19909 df-subrg 19937 df-abv 19992 df-staf 20020 df-srng 20021 df-lmod 20040 df-lss 20109 df-lmhm 20199 df-lvec 20280 df-sra 20349 df-rgmod 20350 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-refld 20722 df-phl 20743 df-dsmm 20849 df-frlm 20864 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-xms 23381 df-ms 23382 df-nm 23644 df-ngp 23645 df-tng 23646 df-nrg 23647 df-nlm 23648 df-clm 24132 df-cph 24237 df-tcph 24238 df-rrx 24454

This theorem is referenced by: rrxngp 43716

W3C validator