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Theorem rrxds 25300

Description: The distance over generalized Euclidean spaces. Compare with df-rrn 37827. (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 25294	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6865	. 2 ⊢ (𝐼 ∈ 𝑉 → (dist‘𝐻) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		resrng 21537	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
5		srngring 20762	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		eqid 2730	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
8	7	frlmlmod 21665	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
9	6, 8	mpan 690	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
10		lmodgrp 20780	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
11		eqid 2730	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
12		eqid 2730	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
13		eqid 2730	. . . 4 ⊢ (-_g‘(ℝ_fld freeLMod 𝐼)) = (-_g‘(ℝ_fld freeLMod 𝐼))
14	11, 12, 13	tcphds 25138	. . 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 2730	. . . . . . . 8 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17	16, 13	grpsubf 18958	. . . . . . 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 25295	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
21		rebase 21522	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
22		re0g 21528	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℝ_fld)
23		eqid 2730	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
24	7, 21, 22, 23	frlmbas 21671	. . . . . . . . . 10 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
25	6, 24	mpan 690	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
26	20, 25	eqtrd 2765	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
27	26	sqxpeqd 5673	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐵 × 𝐵) = ((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼))))
28	27, 26	feq23d 6686	. . . . . 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 7563	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) ∈ 𝐵)
31	29	ffnd 6692	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) Fn (𝐵 × 𝐵))
32		fnov 7523	. . . . 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 25298	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))) = (norm‘𝐻))
35	2	fveq2d 6865	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
36	34, 35	eqtr2d 2766	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))))
37		fveq1 6860	. . . . . . . 8 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (ℎ‘𝑥) = ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥))
38	37	oveq1d 7405	. . . . . . 7 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → ((ℎ‘𝑥)↑2) = (((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥)↑2))
39	38	mpteq2dv 5204	. . . . . 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 7406	. . . . 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 6865	. . . 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 8077	. . 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 1136	. . . . . . . . . . . . . 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 1114	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
48	7, 21, 16	frlmbasmap 21675	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
49	43, 47, 48	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
50		elmapi 8825	. . . . . . . . . . . . 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 6692	. . . . . . . . . . 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 1114	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
56	7, 21, 16	frlmbasmap 21675	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑔 ∈ (Base‘(ℝ_fld freeLMod 𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
57	43, 55, 56	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑_m 𝐼))
58		elmapi 8825	. . . . . . . . . . . . 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 6692	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼)
61		inidm 4193	. . . . . . . . . . 11 ⊢ (𝐼 ∩ 𝐼) = 𝐼
62		eqidd 2731	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
63		eqidd 2731	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥))
64	52, 60, 43, 43, 61, 62, 63	offval 7665	. . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (-_g‘ℝ_fld) = (-_g‘ℝ_fld)
68	7, 16, 65, 66, 46, 54, 67, 13	frlmsubgval 21681	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
69	68	3impb 1114	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑓 ∘_f (-_g‘ℝ_fld)𝑔))
70	51	ffvelcdmda 7059	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
71	59	ffvelcdmda 7059	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
72	67	resubgval 21525	. . . . . . . . . . . 12 ⊢ (((𝑓‘𝑥) ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
73	70, 71, 72	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑥)(-_g‘ℝ_fld)(𝑔‘𝑥)))
74	73	mpteq2dva 5203	. . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) − (𝑔‘𝑥))))
76	70, 71	resubcld 11613	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
77	75, 76	fvmpt2d 6984	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥) = ((𝑓‘𝑥) − (𝑔‘𝑥)))
78	77	oveq1d 7405	. . . . . . 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 5203	. . . . . 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 7406	. . . . 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 6865	. . . 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 7469	. . 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 2765	. 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 2772	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 1086 = wceq 1540 ∈ wcel 2109 {crab 3408 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ∘ ccom 5645 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ∘_f cof 7654 ↑_m cmap 8802 finSupp cfsupp 9319 ℝcr 11074 0cc0 11075 − cmin 11412 2c2 12248 ↑cexp 14033 √csqrt 15206 Basecbs 17186 distcds 17236 Σ_g cgsu 17410 Grpcgrp 18872 -_gcsg 18874 Ringcrg 20149 *-Ringcsr 20754 LModclmod 20773 ℝ_fldcrefld 21520 freeLMod cfrlm 21662 normcnm 24471 toℂPreHilctcph 25074 ℝ^crrx 25290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17411 df-prds 17417 df-pws 17419 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-ghm 19152 df-cmn 19719 df-abl 19720 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 20462 df-subrg 20486 df-drng 20647 df-field 20648 df-staf 20755 df-srng 20756 df-lmod 20775 df-lss 20845 df-sra 21087 df-rgmod 21088 df-cnfld 21272 df-refld 21521 df-dsmm 21648 df-frlm 21663 df-nm 24477 df-tng 24479 df-tcph 25076 df-rrx 25292

This theorem is referenced by: rrxmval 25312 rrxmfval 25313 rrxdsfi 25318 rrxtopn 46289

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