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Theorem rrxds 25552

Description: The distance over generalized Euclidean spaces. Compare with df-rrn 38477. (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 25546	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6885	. 2 ⊢ (𝐼 ∈ 𝑉 → (dist‘𝐻) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		resrng 21771	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
5		srngring 20949	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		eqid 2763	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
8	7	frlmlmod 21899	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
9	6, 8	mpan 702	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
10		lmodgrp 20988	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
11		eqid 2763	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
12		eqid 2763	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
13		eqid 2763	. . . 4 ⊢ (-_g‘(ℝ_fld freeLMod 𝐼)) = (-_g‘(ℝ_fld freeLMod 𝐼))
14	11, 12, 13	tcphds 25390	. . 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 19	. 2 ⊢ (𝐼 ∈ 𝑉 → ((norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) ∘ (-_g‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
16		eqid 2763	. . . . . . . 8 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17	16, 13	grpsubf 19080	. . . . . . 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 19	. . . . . 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 25547	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
21		rebase 21756	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
22		re0g 21762	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℝ_fld)
23		eqid 2763	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
24	7, 21, 22, 23	frlmbas 21905	. . . . . . . . . 10 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
25	6, 24	mpan 702	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
26	20, 25	eqtrd 2798	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
27	26	sqxpeqd 5693	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐵 × 𝐵) = ((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼))))
28	27, 26	feq23d 6700	. . . . . 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 260	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)):(𝐵 × 𝐵)⟶𝐵)
30	29	fovcdmda 7581	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) ∈ 𝐵)
31	29	ffnd 6706	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) Fn (𝐵 × 𝐵))
32		fnov 7541	. . . . 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 221	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)))
34	1, 19	rrxnm 25550	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))) = (norm‘𝐻))
35	2	fveq2d 6885	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
36	34, 35	eqtr2d 2799	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))))
37		fveq1 6880	. . . . . . . 8 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (ℎ‘𝑥) = ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥))
38	37	oveq1d 7425	. . . . . . 7 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → ((ℎ‘𝑥)↑2) = (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))
39	38	mpteq2dv 5205	. . . . . 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 7426	. . . . 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 6885	. . . 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 8086	. . 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 1154	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝑉)
44		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
45	26	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
46	44, 45	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
47	46	3impb 1132	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
48	7, 21, 16	frlmbasmap 21909	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
49	43, 47, 48	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
50		elmapi 8842	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
51	49, 50	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓:𝐼⟶ℝ)
52	51	ffnd 6706	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐼)
53		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
54	53, 45	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
55	54	3impb 1132	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
56	7, 21, 16	frlmbasmap 21909	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
57	43, 55, 56	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑_m 𝐼))
58		elmapi 8842	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℝ)
59	57, 58	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐼⟶ℝ)
60	59	ffnd 6706	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼)
61		inidm 4179	. . . . . . . . . . 11 ⊢ (𝐼 ∩ 𝐼) = 𝐼
62		eqidd 2764	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
63		eqidd 2764	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥))
64	52, 60, 43, 43, 61, 62, 63	offval 7683	. . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ 𝑉)
67		eqid 2763	. . . . . . . . . . . 12 ⊢ (-_g‘ℝ_fld) = (-_g‘ℝ_fld)
68	7, 16, 65, 66, 46, 54, 67, 13	frlmsubgval 21915	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
69	68	3impb 1132	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
70	51	ffvelcdmda 7079	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
71	59	ffvelcdmda 7079	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
72	67	resubgval 21759	. . . . . . . . . . . 12 ⊢ (((𝑓‘𝑥) ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
73	70, 71, 72	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
74	73	mpteq2dva 5204	. . . . . . . . . 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 2808	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) − (𝑔‘𝑥))))
76	70, 71	resubcld 11637	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
77	75, 76	fvmpt2d 7003	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥) = ((𝑓‘𝑥) − (𝑔‘𝑥)))
78	77	oveq1d 7425	. . . . . . 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 5204	. . . . . 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 7426	. . . . 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 6885	. . . 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 7487	. . 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 2798	. 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 2805	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 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 {crab 3416 class class class wbr 5109 ↦ cmpt 5192 × cxp 5659 ∘ ccom 5665 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ∘_f cof 7672 ↑_m cmap 8820 finSupp cfsupp 9317 ℝcr 11094 0cc0 11095 − cmin 11436 2c2 12290 ↑cexp 14093 √csqrt 15280 Basecbs 17264 distcds 17314 Σ_g cgsu 17488 Grpcgrp 18995 -_gcsg 18997 Ringcrg 20310 *-Ringcsr 20941 LModclmod 20981 ℝ_fldcrefld 21754 freeLMod cfrlm 21896 normcnm 24733 toℂPreHilctcph 25326 ℝ^crrx 25542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-rp 13012 df-fz 13531 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-0g 17489 df-prds 17495 df-pws 17497 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-ghm 19279 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-rhm 20550 df-subrng 20645 df-subrg 20669 df-drng 20829 df-field 20830 df-staf 20942 df-srng 20943 df-lmod 20983 df-lss 21053 df-sra 21294 df-rgmod 21295 df-cnfld 21523 df-refld 21755 df-dsmm 21882 df-frlm 21897 df-nm 24739 df-tng 24741 df-tcph 25328 df-rrx 25544

This theorem is referenced by: rrxmval 25564 rrxmfval 25565 rrxdsfi 25570 rrxtopn 46998

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