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Theorem rrxcph 25743

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 25738	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2763	. . 3 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2763	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2763	. . 3 ⊢ (Scalar‘(ℝ_fld freeLMod 𝐼)) = (Scalar‘(ℝ_fld freeLMod 𝐼))
6		eqid 2763	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
7		rebase 21942	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
8		remulr 21947	. . . 4 ⊢ · = (._r‘ℝ_fld)
9		eqid 2763	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
10		eqid 2763	. . . 4 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
11		re0g 21948	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
12		refldcj 21956	. . . 4 ⊢ ∗ = (*_𝑟‘ℝ_fld)
13		refld 21955	. . . . 5 ⊢ ℝ_fld ∈ Field
14	13	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld ∈ Field)
15		fconstmpt 5717	. . . . 5 ⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)
16	6, 7, 4	frlmbasf 22096	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓:𝐼⟶ℝ)
17	16	ffnd 6732	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 Fn 𝐼)
18	17	3adant3 1150	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → 𝑓 Fn 𝐼)
19		simpl 488	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
22	6, 7, 8, 4, 9	frlmipval 22115	. . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . 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 4126	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∩ 𝐼) = 𝐼
25		eqidd 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
26	17, 17, 19, 19, 24, 25, 25	offval 7726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑓‘𝑥))))
27	16	ffvelcdmda 7108	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
28	27, 27	remulcld 11366	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
29	26, 28	fmpt3d 7140	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓):𝐼⟶ℝ)
30		ovexd 7478	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) ∈ V)
31	29	ffund 6736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → Fun (𝑓 ∘_f · 𝑓))
32	6, 11, 4	frlmbasfsupp 22094	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 finSupp 0)
33		0red 11338	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ∈ ℝ)
34		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
35	34	recnd 11364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
36	35	mul02d 11535	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ℝ) → (0 · 𝑥) = 0)
37	19, 33, 16, 16, 36	suppofss1d 8249	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ (𝑓 supp 0))
38		fsuppsssupp 9400	. . . . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 ∘_f · 𝑓) finSupp 0)
40		regsumsupp 21958	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 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 8222	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
43	42, 16	fssdm 6751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ⊆ 𝐼)
44	37, 43	sstrd 3923	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼)
45	44	sselda 3913	. . . . . . . . . . . . . . . . . . 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 7728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → ((𝑓 ∘_f · 𝑓)‘𝑥) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
47	45, 46	syldan 603	. . . . . . . . . . . . . . . . . 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 15898	. . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 9389	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (𝑓 supp 0) ∈ Fin)
55	54, 37	ssfid 9288	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → ((𝑓 ∘_f · 𝑓) supp 0) ∈ Fin)
56	45, 28	syldan 603	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)) → ((𝑓‘𝑥) · (𝑓‘𝑥)) ∈ ℝ)
57	27	msqge0d 11911	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((𝑓‘𝑥) · (𝑓‘𝑥)))
58	45, 57	syldan 603	. . . . . . . . . . . . . . 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 15994	. . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 3258	. . . . . . . . . . 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 728	. . . . . . . . . 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 1187	. . . . . . . . . . . . 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 11364	. . . . . . . . . . . 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 11991	. . . . . . . . . . 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 486	. . . . . . . . . 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 235	. . . . . . . . 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 918	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) = 0 ∨ (𝑓‘𝑥) = 0) ↔ (𝑓‘𝑥) = 0)
70	68, 69	sylib 221	. . . . . . . 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 1150	. . . . . . . . . . 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 486	. . . . . . . . . 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 3936	. . . . . . . . . 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 1210	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑉)
75		0red 11338	. . . . . . . . . 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 8240	. . . . . . . . 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 1187	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . 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 282	. . . . . . . . . . 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 290	. . . . . . . . . 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 482	. . . . . . . . 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 603	. . . . . . . 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 4441	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∘_f · 𝑓) supp 0) ⊆ 𝐼 ↔ (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) = 𝐼)
84	44, 83	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → (((𝑓 ∘_f · 𝑓) supp 0) ∪ (𝐼 ∖ ((𝑓 ∘_f · 𝑓) supp 0))) = 𝐼)
85	84	eleq2d 2850	. . . . . . . . . . 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 1150	. . . . . . . . . 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 483	. . . . . . . . 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 4155	. . . . . . . . 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 221	. . . . . . . 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 973	. . . . . . 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 3158	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0)
92		fconstfv 7241	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0))
93		c0ex 11327	. . . . . . . 8 ⊢ 0 ∈ V
94	93	fconst2 7234	. . . . . . 7 ⊢ (𝑓:𝐼⟶{0} ↔ 𝑓 = (𝐼 × {0}))
95	92, 94	sylbb1 240	. . . . . 6 ⊢ ((𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = 0) → 𝑓 = (𝐼 × {0}))
96	18, 91, 95	syl2anc 596	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ∧ (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = 0) → 𝑓 = (𝐼 × {0}))
97		isfld 21023	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
98	13, 97	mpbi 233	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing)
99	98	simpli 489	. . . . . . . . 9 ⊢ ℝ_fld ∈ DivRing
100		drngring 21017	. . . . . . . . 9 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
101	99, 100	ax-mp 5	. . . . . . . 8 ⊢ ℝ_fld ∈ Ring
102	6, 11	frlm0 22090	. . . . . . . 8 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
103	101, 102	mpan 703	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
104	103, 15	eqtr3di 2813	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = (𝑥 ∈ 𝐼 ↦ 0))
105	104	3ad2ant1 1151	. . . . 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 2824	. . . 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 15335	. . . . 5 ⊢ (𝑥 ∈ ℝ → (∗‘𝑥) = 𝑥)
108	107	adantl 487	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ ℝ) → (∗‘𝑥) = 𝑥)
109		id 23	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
110	6, 7, 8, 4, 9, 10, 11, 12, 14, 106, 108, 109	frlmphl 22117	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ PreHil)
111	6	frlmsca 22089	. . . . 5 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
112	13, 111	mpan 703	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ_fld = (Scalar‘(ℝ_fld freeLMod 𝐼)))
113		df-refld 21941	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
114	112, 113	eqtr3di 2813	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(ℝ_fld freeLMod 𝐼)) = (ℂ_fld ↾_s ℝ))
115		simpr1 1213	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → 𝑓 ∈ ℝ)
116		simpr3 1215	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → 0 ≤ 𝑓)
117	115, 116	resqrtcld 15614	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ ℝ ∧ 𝑓 ∈ ℝ ∧ 0 ≤ 𝑓)) → (√‘𝑓) ∈ ℝ)
118	55, 56, 58	fsumge0 15991	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ Σ𝑥 ∈ ((𝑓 ∘_f · 𝑓) supp 0)((𝑓‘𝑥) · (𝑓‘𝑥)))
119	118, 49	breqtrrd 5136	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 0 ≤ (ℝ_fld Σ_g (𝑓 ∘_f · 𝑓)))
120	119, 23	breqtrrd 5136	. . 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 25588	. 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ ℂPreHil)
122	2, 121	eqeltrd 2864	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∪ cun 4152 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 × cxp 5653 Fun wfun 6552 Fn wfn 6553 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 ∘_f cof 7715 supp csupp 8204 finSupp cfsupp 9381 ℝcr 11226 0cc0 11227 · cmul 11232 ≤ cle 11371 ∗ccj 15292 Σcsu 15882 Basecbs 17416 ↾_s cress 17437 Scalarcsca 17460 ·_𝑖cip 17462 0_gc0g 17639 Σ_g cgsu 17640 Ringcrg 20489 CRingccrg 20490 DivRingcdr 21010 Fieldcfield 21011 ℂ_fldccnfld 21708 ℝ_fldcrefld 21940 freeLMod cfrlm 22082 ℂPreHilccph 25517 toℂPreHilctcph 25518 ℝ^crrx 25734

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306 ax-mulf 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-map 8877 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ico 13509 df-fz 13667 df-fzo 13816 df-seq 14172 df-exp 14232 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-clim 15684 df-sum 15883 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-prds 17647 df-pws 17649 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-subg 19362 df-ghm 19457 df-cntz 19561 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-dvr 20661 df-rhm 20732 df-subrng 20828 df-subrg 20852 df-drng 21012 df-field 21013 df-abv 21096 df-staf 21126 df-srng 21127 df-lmod 21167 df-lss 21237 df-lmhm 21327 df-lvec 21408 df-sra 21478 df-rgmod 21479 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-cnfld 21709 df-refld 21941 df-phl 21962 df-dsmm 22068 df-frlm 22083 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-xms 24669 df-ms 24670 df-nm 24931 df-ngp 24932 df-tng 24933 df-nrg 24934 df-nlm 24935 df-clm 25414 df-cph 25519 df-tcph 25520 df-rrx 25736

This theorem is used by: rrxngp 47500

W3C validator