Theorem nmpropd2 24448

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 2766	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		nmpropd2.5	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
5	1	sqxpeqd 5699	. . . . . . 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 2766	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	2	sqxpeqd 5699	. . . . . 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 2766	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11		eqidd 2725	. . . 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 18591	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
14	10, 11, 13	oveq123d 7423	. . 3 ⊢ (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿)))
15	3, 14	mpteq12dv 5230	. 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 2724	. . . 4 ⊢ (norm‘𝐾) = (norm‘𝐾)
18		eqid 2724	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2724	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2724	. . . 4 ⊢ (dist‘𝐾) = (dist‘𝐾)
21		eqid 2724	. . . 4 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
22	17, 18, 19, 20, 21	nmfval2 24444	. . 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 18877	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
25	16, 24	mpbid 231	. . 3 ⊢ (𝜑 → 𝐿 ∈ Grp)
26		eqid 2724	. . . 4 ⊢ (norm‘𝐿) = (norm‘𝐿)
27		eqid 2724	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
28		eqid 2724	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
29		eqid 2724	. . . 4 ⊢ (dist‘𝐿) = (dist‘𝐿)
30		eqid 2724	. . . 4 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
31	26, 27, 28, 29, 30	nmfval2 24444	. . 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 2774	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5222   × cxp 5665   ↾ cres 5669  ‘cfv 6534  (class class class)co 7402  Basecbs 17149  +_gcplusg 17202  distcds 17211  0_gc0g 17390  Grpcgrp 18859  normcnm 24429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-riota 7358  df-ov 7405  df-0g 17392  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18862  df-nm 24435

This theorem is referenced by:  ngppropd  24490