Theorem nmpropd2 23657

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 2780	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		nmpropd2.5	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
5	1	sqxpeqd 5612	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
6	5	reseq2d 5880	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
7	4, 6	eqtr3d 2780	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	2	sqxpeqd 5612	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
9	8	reseq2d 5880	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
10	7, 9	eqtr3d 2780	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11		eqidd 2739	. . . 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 18261	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
14	10, 11, 13	oveq123d 7276	. . 3 ⊢ (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿)))
15	3, 14	mpteq12dv 5161	. 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 2738	. . . 4 ⊢ (norm‘𝐾) = (norm‘𝐾)
18		eqid 2738	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2738	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2738	. . . 4 ⊢ (dist‘𝐾) = (dist‘𝐾)
21		eqid 2738	. . . 4 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
22	17, 18, 19, 20, 21	nmfval2 23653	. . 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 18509	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
25	16, 24	mpbid 231	. . 3 ⊢ (𝜑 → 𝐿 ∈ Grp)
26		eqid 2738	. . . 4 ⊢ (norm‘𝐿) = (norm‘𝐿)
27		eqid 2738	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
28		eqid 2738	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
29		eqid 2738	. . . 4 ⊢ (dist‘𝐿) = (dist‘𝐿)
30		eqid 2738	. . . 4 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
31	26, 27, 28, 29, 30	nmfval2 23653	. . 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 2788	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ↦ cmpt 5153   × cxp 5578   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  distcds 16897  0_gc0g 17067  Grpcgrp 18492  normcnm 23638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-nm 23644

This theorem is referenced by:  ngppropd  23699