Theorem nmpropd 22208

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 2611	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
4		eqidd 2611	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5		nmpropd.2	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))
6	5	oveqdr 6573	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	4, 1, 6	grpidpropd 17084	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
8	2, 3, 7	oveq123d 6570	. . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿)))
9	1, 8	mpteq12dv 4663	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))))
10		eqid 2610	. . 3 ⊢ (norm‘𝐾) = (norm‘𝐾)
11		eqid 2610	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2610	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
13		eqid 2610	. . 3 ⊢ (dist‘𝐾) = (dist‘𝐾)
14	10, 11, 12, 13	nmfval 22203	. 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾)))
15		eqid 2610	. . 3 ⊢ (norm‘𝐿) = (norm‘𝐿)
16		eqid 2610	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
17		eqid 2610	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
18		eqid 2610	. . 3 ⊢ (dist‘𝐿) = (dist‘𝐿)
19	15, 16, 17, 18	nmfval 22203	. 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))
20	9, 14, 19	3eqtr4g 2669	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  distcds 15777  0_gc0g 15923  normcnm 22191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-0g 15925  df-nm 22197

This theorem is referenced by:  sranlm  22298  rlmnm  22303  zlmnm  29338