rrxds - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rrxds		Structured version Visualization version GIF version

Theorem rrxds 25361

Description: The distance over generalized Euclidean spaces. Compare with df-rrn 38077. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)

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

**Assertion**
Ref	Expression
rrxds	⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘𝐻))

Distinct variable groups: 𝑓,𝑔,𝑥,𝐵 𝑓,𝐼,𝑔,𝑥 𝑓,𝑉,𝑔,𝑥 Allowed substitution hints: 𝐻(𝑥,𝑓,𝑔)

Proof of Theorem rrxds Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxval.r	. . . 4 ⊢ 𝐻 = (ℝ^‘𝐼)
2	1	rrxval 25355	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6846	. 2 ⊢ (𝐼 ∈ 𝑉 → (dist‘𝐻) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		resrng 21588	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
5		srngring 20791	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		eqid 2737	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
8	7	frlmlmod 21716	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
9	6, 8	mpan 691	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
10		lmodgrp 20830	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
11		eqid 2737	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
12		eqid 2737	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
13		eqid 2737	. . . 4 ⊢ (-_g‘(ℝ_fld freeLMod 𝐼)) = (-_g‘(ℝ_fld freeLMod 𝐼))
14	11, 12, 13	tcphds 25199	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ Grp → ((norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) ∘ (-_g‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
15	9, 10, 14	3syl 18	. 2 ⊢ (𝐼 ∈ 𝑉 → ((norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) ∘ (-_g‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
16		eqid 2737	. . . . . . . 8 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17	16, 13	grpsubf 18961	. . . . . . 7 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ Grp → (-_g‘(ℝ_fld freeLMod 𝐼)):((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼)))⟶(Base‘(ℝ_fld freeLMod 𝐼)))
18	9, 10, 17	3syl 18	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)):((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼)))⟶(Base‘(ℝ_fld freeLMod 𝐼)))
19		rrxbase.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐻)
20	1, 19	rrxbase 25356	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
21		rebase 21573	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
22		re0g 21579	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℝ_fld)
23		eqid 2737	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
24	7, 21, 22, 23	frlmbas 21722	. . . . . . . . . 10 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
25	6, 24	mpan 691	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
26	20, 25	eqtrd 2772	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
27	26	sqxpeqd 5664	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐵 × 𝐵) = ((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼))))
28	27, 26	feq23d 6665	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((-_g‘(ℝ_fld freeLMod 𝐼)):(𝐵 × 𝐵)⟶𝐵 ↔ (-_g‘(ℝ_fld freeLMod 𝐼)):((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼)))⟶(Base‘(ℝ_fld freeLMod 𝐼))))
29	18, 28	mpbird 257	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)):(𝐵 × 𝐵)⟶𝐵)
30	29	fovcdmda 7539	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) ∈ 𝐵)
31	29	ffnd 6671	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) Fn (𝐵 × 𝐵))
32		fnov 7499	. . . . 5 ⊢ ((-_g‘(ℝ_fld freeLMod 𝐼)) Fn (𝐵 × 𝐵) ↔ (-_g‘(ℝ_fld freeLMod 𝐼)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)))
33	31, 32	sylib 218	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)))
34	1, 19	rrxnm 25359	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))) = (norm‘𝐻))
35	2	fveq2d 6846	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
36	34, 35	eqtr2d 2773	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))))
37		fveq1 6841	. . . . . . . 8 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (ℎ‘𝑥) = ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥))
38	37	oveq1d 7383	. . . . . . 7 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → ((ℎ‘𝑥)↑2) = (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))
39	38	mpteq2dv 5194	. . . . . 6 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2)))
40	39	oveq2d 7384	. . . . 5 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))))
41	40	fveq2d 6846	. . . 4 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2)))) = (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2)))))
42	30, 33, 36, 41	fmpoco 8047	. . 3 ⊢ (𝐼 ∈ 𝑉 → ((norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) ∘ (-_g‘(ℝ_fld freeLMod 𝐼))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))))))
43		simp1 1137	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝑉)
44		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
45	26	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
46	44, 45	eleqtrd 2839	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
47	46	3impb 1115	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
48	7, 21, 16	frlmbasmap 21726	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
49	43, 47, 48	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
50		elmapi 8798	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓:𝐼⟶ℝ)
52	51	ffnd 6671	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐼)
53		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
54	53, 45	eleqtrd 2839	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
55	54	3impb 1115	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
56	7, 21, 16	frlmbasmap 21726	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
57	43, 55, 56	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑_m 𝐼))
58		elmapi 8798	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℝ)
59	57, 58	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐼⟶ℝ)
60	59	ffnd 6671	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼)
61		inidm 4181	. . . . . . . . . . 11 ⊢ (𝐼 ∩ 𝐼) = 𝐼
62		eqidd 2738	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
63		eqidd 2738	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥))
64	52, 60, 43, 43, 61, 62, 63	offval 7641	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓 ∘_f (-_g‘ℝ_fld)𝑔) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥))))
65	6	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ℝ_fld ∈ Ring)
66		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ 𝑉)
67		eqid 2737	. . . . . . . . . . . 12 ⊢ (-_g‘ℝ_fld) = (-_g‘ℝ_fld)
68	7, 16, 65, 66, 46, 54, 67, 13	frlmsubgval 21732	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
69	68	3impb 1115	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
70	51	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
71	59	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
72	67	resubgval 21576	. . . . . . . . . . . 12 ⊢ (((𝑓‘𝑥) ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
73	70, 71, 72	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
74	73	mpteq2dva 5193	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) − (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥))))
75	64, 69, 74	3eqtr4d 2782	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) − (𝑔‘𝑥))))
76	70, 71	resubcld 11577	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
77	75, 76	fvmpt2d 6963	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥) = ((𝑓‘𝑥) − (𝑔‘𝑥)))
78	77	oveq1d 7383	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2) = (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))
79	78	mpteq2dva 5193	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))
80	79	oveq2d 7384	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))
81	80	fveq2d 6846	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2)))) = (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))
82	81	mpoeq3dva 7445	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
83	42, 82	eqtrd 2772	. 2 ⊢ (𝐼 ∈ 𝑉 → ((norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) ∘ (-_g‘(ℝ_fld freeLMod 𝐼))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
84	3, 15, 83	3eqtr2rd 2779	1 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3401 class class class wbr 5100 ↦ cmpt 5181 × cxp 5630 ∘ ccom 5636 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ∘_f cof 7630 ↑_m cmap 8775 finSupp cfsupp 9276 ℝcr 11037 0cc0 11038 − cmin 11376 2c2 12212 ↑cexp 13996 √csqrt 15168 Basecbs 17148 distcds 17198 Σ_g cgsu 17372 Grpcgrp 18875 -_gcsg 18877 Ringcrg 20180 *-Ringcsr 20783 LModclmod 20823 ℝ_fldcrefld 21571 freeLMod cfrlm 21713 normcnm 24532 toℂPreHilctcph 25135 ℝ^crrx 25351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-rp 12918 df-fz 13436 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-prds 17379 df-pws 17381 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-ghm 19154 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-rhm 20420 df-subrng 20491 df-subrg 20515 df-drng 20676 df-field 20677 df-staf 20784 df-srng 20785 df-lmod 20825 df-lss 20895 df-sra 21137 df-rgmod 21138 df-cnfld 21322 df-refld 21572 df-dsmm 21699 df-frlm 21714 df-nm 24538 df-tng 24540 df-tcph 25137 df-rrx 25353

This theorem is referenced by: rrxmval 25373 rrxmfval 25374 rrxdsfi 25379 rrxtopn 46642

W3C validator