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Theorem rrxds 25334

Description: The distance over generalized Euclidean spaces. Compare with df-rrn 37299. (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 25328	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6901	. 2 ⊢ (𝐼 ∈ 𝑉 → (dist‘𝐻) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		resrng 21553	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
5		srngring 20732	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		eqid 2728	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
8	7	frlmlmod 21683	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
9	6, 8	mpan 689	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
10		lmodgrp 20750	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
11		eqid 2728	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
12		eqid 2728	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
13		eqid 2728	. . . 4 ⊢ (-_g‘(ℝ_fld freeLMod 𝐼)) = (-_g‘(ℝ_fld freeLMod 𝐼))
14	11, 12, 13	tcphds 25172	. . 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 2728	. . . . . . . 8 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17	16, 13	grpsubf 18975	. . . . . . 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 25329	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
21		rebase 21538	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
22		re0g 21544	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℝ_fld)
23		eqid 2728	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
24	7, 21, 22, 23	frlmbas 21689	. . . . . . . . . 10 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
25	6, 24	mpan 689	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
26	20, 25	eqtrd 2768	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
27	26	sqxpeqd 5710	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐵 × 𝐵) = ((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼))))
28	27, 26	feq23d 6717	. . . . . 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 7592	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) ∈ 𝐵)
31	29	ffnd 6723	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) Fn (𝐵 × 𝐵))
32		fnov 7552	. . . . 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 217	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)))
34	1, 19	rrxnm 25332	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))) = (norm‘𝐻))
35	2	fveq2d 6901	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
36	34, 35	eqtr2d 2769	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))))
37		fveq1 6896	. . . . . . . 8 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (ℎ‘𝑥) = ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥))
38	37	oveq1d 7435	. . . . . . 7 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → ((ℎ‘𝑥)↑2) = (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))
39	38	mpteq2dv 5250	. . . . . 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 7436	. . . . 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 6901	. . . 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 8100	. . 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 1134	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝑉)
44		simprl 770	. . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
47	46	3impb 1113	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
48	7, 21, 16	frlmbasmap 21693	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
49	43, 47, 48	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
50		elmapi 8868	. . . . . . . . . . . . 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 6723	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐼)
53		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
54	53, 45	eleqtrd 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
55	54	3impb 1113	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
56	7, 21, 16	frlmbasmap 21693	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
57	43, 55, 56	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑_m 𝐼))
58		elmapi 8868	. . . . . . . . . . . . 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 6723	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼)
61		inidm 4219	. . . . . . . . . . 11 ⊢ (𝐼 ∩ 𝐼) = 𝐼
62		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
63		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥))
64	52, 60, 43, 43, 61, 62, 63	offval 7694	. . . . . . . . . 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 2728	. . . . . . . . . . . 12 ⊢ (-_g‘ℝ_fld) = (-_g‘ℝ_fld)
68	7, 16, 65, 66, 46, 54, 67, 13	frlmsubgval 21699	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
69	68	3impb 1113	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
70	51	ffvelcdmda 7094	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
71	59	ffvelcdmda 7094	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
72	67	resubgval 21541	. . . . . . . . . . . 12 ⊢ (((𝑓‘𝑥) ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
73	70, 71, 72	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
74	73	mpteq2dva 5248	. . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) − (𝑔‘𝑥))))
76	70, 71	resubcld 11673	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
77	75, 76	fvmpt2d 7018	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥) = ((𝑓‘𝑥) − (𝑔‘𝑥)))
78	77	oveq1d 7435	. . . . . . 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 5248	. . . . . 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 7436	. . . . 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 6901	. . . 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 7497	. . 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 2768	. 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 2775	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 1085 = wceq 1534 ∈ wcel 2099 {crab 3429 class class class wbr 5148 ↦ cmpt 5231 × cxp 5676 ∘ ccom 5682 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ∘_f cof 7683 ↑_m cmap 8845 finSupp cfsupp 9386 ℝcr 11138 0cc0 11139 − cmin 11475 2c2 12298 ↑cexp 14059 √csqrt 15213 Basecbs 17180 distcds 17242 Σ_g cgsu 17422 Grpcgrp 18890 -_gcsg 18892 Ringcrg 20173 *-Ringcsr 20724 LModclmod 20743 ℝ_fldcrefld 21536 freeLMod cfrlm 21680 normcnm 24498 toℂPreHilctcph 25108 ℝ^crrx 25324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-fz 13518 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-prds 17429 df-pws 17431 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-ghm 19168 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-drng 20626 df-field 20627 df-staf 20725 df-srng 20726 df-lmod 20745 df-lss 20816 df-sra 21058 df-rgmod 21059 df-cnfld 21280 df-refld 21537 df-dsmm 21666 df-frlm 21681 df-nm 24504 df-tng 24506 df-tcph 25110 df-rrx 25326

This theorem is referenced by: rrxmval 25346 rrxmfval 25347 rrxdsfi 25352 rrxtopn 45672

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