Theorem nmpropd2 24805

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 2802	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		nmpropd2.5	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
5	1	sqxpeqd 5695	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
6	5	reseq2d 5980	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
7	4, 6	eqtr3d 2802	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	2	sqxpeqd 5695	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
9	8	reseq2d 5980	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
10	7, 9	eqtr3d 2802	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11		eqidd 2766	. . . 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 18747	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
14	10, 11, 13	oveq123d 7440	. . 3 ⊢ (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g‘𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g‘𝐿)))
15	3, 14	mpteq12dv 5200	. 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 2765	. . . 4 ⊢ (norm‘𝐾) = (norm‘𝐾)
18		eqid 2765	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2765	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2765	. . . 4 ⊢ (dist‘𝐾) = (dist‘𝐾)
21		eqid 2765	. . . 4 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
22	17, 18, 19, 20, 21	nmfval2 24801	. . 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 19064	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
25	16, 24	mpbid 235	. . 3 ⊢ (𝜑 → 𝐿 ∈ Grp)
26		eqid 2765	. . . 4 ⊢ (norm‘𝐿) = (norm‘𝐿)
27		eqid 2765	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
28		eqid 2765	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
29		eqid 2765	. . . 4 ⊢ (dist‘𝐿) = (dist‘𝐿)
30		eqid 2765	. . . 4 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
31	26, 27, 28, 29, 30	nmfval2 24801	. . 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 2810	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5194   × cxp 5661   ↾ cres 5665  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  +_gcplusg 17334  distcds 17343  0_gc0g 17516  Grpcgrp 19046  normcnm 24786

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7376  df-ov 7422  df-0g 17518  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-grp 19049  df-nm 24792

This theorem is used by:  ngppropd  24847