Theorem nmpropd2 23204

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 2858	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		nmpropd2.5	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
5	1	sqxpeqd 5587	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
6	5	reseq2d 5853	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
7	4, 6	eqtr3d 2858	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	2	sqxpeqd 5587	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
9	8	reseq2d 5853	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
10	7, 9	eqtr3d 2858	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11		eqidd 2822	. . . 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 17872	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
14	10, 11, 13	oveq123d 7177	. . 3 ⊢ (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿)))
15	3, 14	mpteq12dv 5151	. 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 2821	. . . 4 ⊢ (norm‘𝐾) = (norm‘𝐾)
18		eqid 2821	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2821	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2821	. . . 4 ⊢ (dist‘𝐾) = (dist‘𝐾)
21		eqid 2821	. . . 4 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
22	17, 18, 19, 20, 21	nmfval2 23200	. . 3 ⊢ (𝐾 ∈ Grp → (norm‘𝐾) = (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾))))
23	16, 22	syl 17	. 2 ⊢ (𝜑 → (norm‘𝐾) = (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾))))
24	1, 2, 12	grppropd 18118	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
25	16, 24	mpbid 234	. . 3 ⊢ (𝜑 → 𝐿 ∈ Grp)
26		eqid 2821	. . . 4 ⊢ (norm‘𝐿) = (norm‘𝐿)
27		eqid 2821	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
28		eqid 2821	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
29		eqid 2821	. . . 4 ⊢ (dist‘𝐿) = (dist‘𝐿)
30		eqid 2821	. . . 4 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
31	26, 27, 28, 29, 30	nmfval2 23200	. . 3 ⊢ (𝐿 ∈ Grp → (norm‘𝐿) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿))))
32	25, 31	syl 17	. 2 ⊢ (𝜑 → (norm‘𝐿) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿))))
33	15, 23, 32	3eqtr4d 2866	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5146   × cxp 5553   ↾ cres 5557  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  distcds 16574  0_gc0g 16713  Grpcgrp 18103  normcnm 23186

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-riota 7114  df-ov 7159  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-nm 23192

This theorem is referenced by:  ngppropd  23246