Theorem nmpropd 22806

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

Hypotheses

Ref	Expression
nmpropd.1	⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
nmpropd.2	⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))
nmpropd.3	⊢ (𝜑 → (dist‘𝐾) = (dist‘𝐿))

Assertion

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

Proof of Theorem nmpropd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmpropd.1	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
2		nmpropd.3	. . . 4 ⊢ (𝜑 → (dist‘𝐾) = (dist‘𝐿))
3		eqidd 2778	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
4		eqidd 2778	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5		nmpropd.2	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))
6	5	oveqdr 6950	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	4, 1, 6	grpidpropd 17647	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
8	2, 3, 7	oveq123d 6943	. . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿)))
9	1, 8	mpteq12dv 4969	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))))
10		eqid 2777	. . 3 ⊢ (norm‘𝐾) = (norm‘𝐾)
11		eqid 2777	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2777	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
13		eqid 2777	. . 3 ⊢ (dist‘𝐾) = (dist‘𝐾)
14	10, 11, 12, 13	nmfval 22801	. 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾)))
15		eqid 2777	. . 3 ⊢ (norm‘𝐿) = (norm‘𝐿)
16		eqid 2777	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
17		eqid 2777	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
18		eqid 2777	. . 3 ⊢ (dist‘𝐿) = (dist‘𝐿)
19	15, 16, 17, 18	nmfval 22801	. 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))
20	9, 14, 19	3eqtr4g 2838	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106   ↦ cmpt 4965  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +_gcplusg 16338  distcds 16347  0_gc0g 16486  normcnm 22789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-0g 16488  df-nm 22795

This theorem is referenced by:  sranlm  22896  rlmnm  22901  zlmnm  30608