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Theorem rrxds 25241

Description: The distance over generalized Euclidean spaces. Compare with df-rrn 37158. (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 25235	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6895	. 2 ⊢ (𝐼 ∈ 𝑉 → (dist‘𝐻) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		resrng 21484	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
5		srngring 20691	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		eqid 2731	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
8	7	frlmlmod 21614	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
9	6, 8	mpan 687	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
10		lmodgrp 20709	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
11		eqid 2731	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
12		eqid 2731	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
13		eqid 2731	. . . 4 ⊢ (-_g‘(ℝ_fld freeLMod 𝐼)) = (-_g‘(ℝ_fld freeLMod 𝐼))
14	11, 12, 13	tcphds 25079	. . 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 2731	. . . . . . . 8 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17	16, 13	grpsubf 18945	. . . . . . 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 25236	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
21		rebase 21469	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
22		re0g 21475	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℝ_fld)
23		eqid 2731	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
24	7, 21, 22, 23	frlmbas 21620	. . . . . . . . . 10 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
25	6, 24	mpan 687	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
26	20, 25	eqtrd 2771	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
27	26	sqxpeqd 5708	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝐵 × 𝐵) = ((Base‘(ℝ_fld freeLMod 𝐼)) × (Base‘(ℝ_fld freeLMod 𝐼))))
28	27, 26	feq23d 6712	. . . . . 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 7582	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) ∈ 𝐵)
31	29	ffnd 6718	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (-_g‘(ℝ_fld freeLMod 𝐼)) Fn (𝐵 × 𝐵))
32		fnov 7543	. . . . 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 25239	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))) = (norm‘𝐻))
35	2	fveq2d 6895	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
36	34, 35	eqtr2d 2772	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (ℎ ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)↑2))))))
37		fveq1 6890	. . . . . . . 8 ⊢ (ℎ = (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) → (ℎ‘𝑥) = ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥))
38	37	oveq1d 7427	. . . . . . 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 7428	. . . . 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 6895	. . . 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 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝑉)
44		simprl 768	. . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . 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 21624	. . . . . . . . . . . . . 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 8849	. . . . . . . . . . . . 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 6718	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐼)
53		simprr 770	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
54	53, 45	eleqtrd 2834	. . . . . . . . . . . . . . 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 21624	. . . . . . . . . . . . . 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 8849	. . . . . . . . . . . . 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 6718	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼)
61		inidm 4218	. . . . . . . . . . 11 ⊢ (𝐼 ∩ 𝐼) = 𝐼
62		eqidd 2732	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) = (𝑓‘𝑥))
63		eqidd 2732	. . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ 𝑉)
67		eqid 2731	. . . . . . . . . . . 12 ⊢ (-_g‘ℝ_fld) = (-_g‘ℝ_fld)
68	7, 16, 65, 66, 46, 54, 67, 13	frlmsubgval 21630	. . . . . . . . . . 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 7086	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
71	59	ffvelcdmda 7086	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
72	67	resubgval 21472	. . . . . . . . . . . 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 2781	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) − (𝑔‘𝑥))))
76	70, 71	resubcld 11649	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
77	75, 76	fvmpt2d 7011	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓(-_g‘(ℝ_fld freeLMod 𝐼))𝑔)‘𝑥) = ((𝑓‘𝑥) − (𝑔‘𝑥)))
78	77	oveq1d 7427	. . . . . . 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 7428	. . . . 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 6895	. . . 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 7489	. . 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 2771	. 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 2778	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 2105 {crab 3431 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ∘_f cof 7672 ↑_m cmap 8826 finSupp cfsupp 9367 ℝcr 11115 0cc0 11116 − cmin 11451 2c2 12274 ↑cexp 14034 √csqrt 15187 Basecbs 17151 distcds 17213 Σ_g cgsu 17393 Grpcgrp 18861 -_gcsg 18863 Ringcrg 20134 *-Ringcsr 20683 LModclmod 20702 ℝ_fldcrefld 21467 freeLMod cfrlm 21611 normcnm 24405 toℂPreHilctcph 25015 ℝ^crrx 25231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-staf 20684 df-srng 20685 df-lmod 20704 df-lss 20775 df-sra 21019 df-rgmod 21020 df-cnfld 21234 df-refld 21468 df-dsmm 21597 df-frlm 21612 df-nm 24411 df-tng 24413 df-tcph 25017 df-rrx 25233

This theorem is referenced by: rrxmval 25253 rrxmfval 25254 rrxdsfi 25259 rrxtopn 45459

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