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Theorem rrxcph 25317

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 25312	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2731	. . 3 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2731	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2731	. . 3 ⊢ (Scalar‘(ℝ_fld freeLMod 𝐼)) = (Scalar‘(ℝ_fld freeLMod 𝐼))
6		eqid 2731	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
7		rebase 21541	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
8		remulr 21546	. . . 4 ⊢ · = (._r‘ℝ_fld)
9		eqid 2731	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
10		eqid 2731	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
11		re0g 21547	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
12		refldcj 21555	. . . 4 ⊢ ∗ = (*_𝑟‘ℝ_fld)
13		refld 21554	. . . . 5 ⊢ ℝ_fld ∈ Field
14	13	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld ∈ Field)
15		fconstmpt 5678	. . . . 5 ⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)
16	6, 7, 4	frlmbasf 21695	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓:𝐼⟶ℝ)
17	16	ffnd 6652	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 Fn 𝐼)
18	17	3adant3 1132	. . . . . 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 21714	. . . . . . . . . . . . . . . . 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 4177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∩ 𝐼) = 𝐼
25		eqidd 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
26	17, 17, 19, 19, 24, 25, 25	offval 7619	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑓‘𝑥))))
27	16	ffvelcdmda 7017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
28	27, 27	remulcld 11139	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
29	26, 28	fmpt3d 7049	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓):𝐼⟶ℝ)
30		ovexd 7381	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) ∈ V)
31	29	ffund 6655	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → Fun (𝑓 ∘_f · 𝑓))
32	6, 11, 4	frlmbasfsupp 21693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 finSupp 0)
33		0red 11112	. . . . . . . . . . . . . . . . . . . 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 11137	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
36	35	mul02d 11308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → (0 · 𝑥) = 0)
37	19, 33, 16, 16, 36	suppofss1d 8134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ (𝑓 supp 0))
38		fsuppsssupp 9265	. . . . . . . . . . . . . . . . . . 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 21557	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 8107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
43	42, 16	fssdm 6670	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ⊆ 𝐼)
44	37, 43	sstrd 3945	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼)
45	44	sselda 3934	. . . . . . . . . . . . . . . . . . 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 7621	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓 ∘_f · 𝑓)‘𝑥) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
47	45, 46	syldan 591	. . . . . . . . . . . . . . . . . 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 15606	. . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 9253	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ∈ Fin)
55	54, 37	ssfid 9153	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ∈ Fin)
56	45, 28	syldan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
57	27	msqge0d 11682	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((𝑓‘𝑥) · (𝑓‘𝑥)))
58	45, 57	syldan 591	. . . . . . . . . . . . . . 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 15702	. . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 3224	. . . . . . . . . . 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 715	. . . . . . . . . 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 1169	. . . . . . . . . . . . 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 11137	. . . . . . . . . . . 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 11762	. . . . . . . . . . 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 232	. . . . . . . . 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 904	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0) ↔ (𝑓‘𝑥) = 0)
70	68, 69	sylib 218	. . . . . . . 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 1132	. . . . . . . . . . 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 3958	. . . . . . . . . 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 1192	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑉)
75		0red 11112	. . . . . . . . . 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 8125	. . . . . . . . 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 1169	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . 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 591	. . . . . . . 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 4432	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼 ↔ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) = 𝐼)
84	44, 83	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) = 𝐼)
85	84	eleq2d 2817	. . . . . . . . . . 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 1132	. . . . . . . . . 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 4103	. . . . . . . . 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 218	. . . . . . . 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 960	. . . . . . 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 3124	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0)
92		fconstfv 7146	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0))
93		c0ex 11103	. . . . . . . 8 ⊢ 0 ∈ V
94	93	fconst2 7139	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ 𝑓 = (𝐼 × {0}))
95	92, 94	sylbb1 237	. . . . . 6 ⊢ ((𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0) → 𝑓 = (𝐼 × {0}))
96	18, 91, 95	syl2anc 584	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → 𝑓 = (𝐼 × {0}))
97		isfld 20653	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
98	13, 97	mpbi 230	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing)
99	98	simpli 483	. . . . . . . . 9 ⊢ ℝ_fld ∈ DivRing
100		drngring 20649	. . . . . . . . 9 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
101	99, 100	ax-mp 5	. . . . . . . 8 ⊢ ℝ_fld ∈ Ring
102	6, 11	frlm0 21689	. . . . . . . 8 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
103	101, 102	mpan 690	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
104	103, 15	eqtr3di 2781	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = (𝑥 ∈ 𝐼 ↦ 0))
105	104	3ad2ant1 1133	. . . . 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 2792	. . . 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 15043	. . . . 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 21716	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ PreHil)
111	6	frlmsca 21688	. . . . 5 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
112	13, 111	mpan 690	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
113		df-refld 21540	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
114	112, 113	eqtr3di 2781	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(ℝ_fld freeLMod 𝐼)) = (ℂ_fld ↾_s ℝ))
115		simpr1 1195	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → 𝑓 ∈ ℝ)
116		simpr3 1197	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → 0 ≤ 𝑓)
117	115, 116	resqrtcld 15322	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → (√‘𝑓) ∈ ℝ)
118	55, 56, 58	fsumge0 15699	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
119	118, 49	breqtrrd 5119	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)))
120	119, 23	breqtrrd 5119	. . 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 25162	. 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ ℂPreHil)
122	2, 121	eqeltrd 2831	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ∖ cdif 3899 ∪ cun 3900 ⊆ wss 3902 {csn 4576 class class class wbr 5091 ↦ cmpt 5172 × cxp 5614 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 supp csupp 8090 finSupp cfsupp 9245 ℝcr 11002 0cc0 11003 · cmul 11008 ≤ cle 11144 ∗ccj 15000 Σcsu 15590 Basecbs 17117 ↾_s cress 17138 Scalarcsca 17161 ·_𝑖cip 17163 0_gc0g 17340 Σ_g cgsu 17341 Ringcrg 20149 CRingccrg 20150 DivRingcdr 20642 Fieldcfield 20643 ℂ_fldccnfld 21289 ℝ_fldcrefld 21539 freeLMod cfrlm 21681 ℂPreHilccph 25091 toℂPreHilctcph 25092 ℝ^crrx 25308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082 ax-mulf 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ico 13248 df-fz 13405 df-fzo 13552 df-seq 13906 df-exp 13966 df-hash 14235 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-clim 15392 df-sum 15591 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-prds 17348 df-pws 17350 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-mhm 18688 df-submnd 18689 df-grp 18846 df-minusg 18847 df-sbg 18848 df-subg 19033 df-ghm 19123 df-cntz 19227 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20459 df-subrg 20483 df-drng 20644 df-field 20645 df-abv 20722 df-staf 20752 df-srng 20753 df-lmod 20793 df-lss 20863 df-lmhm 20954 df-lvec 21035 df-sra 21105 df-rgmod 21106 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-cnfld 21290 df-refld 21540 df-phl 21561 df-dsmm 21667 df-frlm 21682 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-xms 24233 df-ms 24234 df-nm 24495 df-ngp 24496 df-tng 24497 df-nrg 24498 df-nlm 24499 df-clm 24988 df-cph 25093 df-tcph 25094 df-rrx 25310

This theorem is referenced by: rrxngp 46322

W3C validator