Theorem nmpropd2 24822

Description: Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
nmpropd2.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
nmpropd2.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
nmpropd2.3	⊢ (𝜑 → 𝐾 ∈ Grp)
nmpropd2.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
nmpropd2.5	⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
nmpropd2	⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem nmpropd2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmpropd2.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		nmpropd2.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3	1, 2	eqtr3d 2797	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		nmpropd2.5	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
5	1	sqxpeqd 5687	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
6	5	reseq2d 5972	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
7	4, 6	eqtr3d 2797	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	2	sqxpeqd 5687	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
9	8	reseq2d 5972	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
10	7, 9	eqtr3d 2797	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11		eqidd 2761	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
12		nmpropd2.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
13	1, 2, 12	grpidpropd 18756	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
14	10, 11, 13	oveq123d 7435	. . 3 ⊢ (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿)))
15	3, 14	mpteq12dv 5192	. 2 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾))) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿))))
16		nmpropd2.3	. . 3 ⊢ (𝜑 → 𝐾 ∈ Grp)
17		eqid 2760	. . . 4 ⊢ (norm‘𝐾) = (norm‘𝐾)
18		eqid 2760	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2760	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2760	. . . 4 ⊢ (dist‘𝐾) = (dist‘𝐾)
21		eqid 2760	. . . 4 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
22	17, 18, 19, 20, 21	nmfval2 24818	. . 3 ⊢ (𝐾 ∈ Grp → (norm‘𝐾) = (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾))))
23	16, 22	syl 18	. 2 ⊢ (𝜑 → (norm‘𝐾) = (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾))))
24	1, 2, 12	grppropd 19076	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
25	16, 24	mpbid 235	. . 3 ⊢ (𝜑 → 𝐿 ∈ Grp)
26		eqid 2760	. . . 4 ⊢ (norm‘𝐿) = (norm‘𝐿)
27		eqid 2760	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
28		eqid 2760	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
29		eqid 2760	. . . 4 ⊢ (dist‘𝐿) = (dist‘𝐿)
30		eqid 2760	. . . 4 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
31	26, 27, 28, 29, 30	nmfval2 24818	. . 3 ⊢ (𝐿 ∈ Grp → (norm‘𝐿) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿))))
32	25, 31	syl 18	. 2 ⊢ (𝜑 → (norm‘𝐿) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿))))
33	15, 23, 32	3eqtr4d 2805	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5186   × cxp 5653   ↾ cres 5657  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  +_gcplusg 17343  distcds 17352  0_gc0g 17525  Grpcgrp 19058  normcnm 24803

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-riota 7371  df-ov 7417  df-0g 17527  df-mgm 18731  df-sgrp 18822  df-mnd 18838  df-grp 19061  df-nm 24809

This theorem is used by:  ngppropd  24864