Theorem nmpropd 24720

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 2770	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
4		eqidd 2770	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5		nmpropd.2	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))
6	5	oveqdr 7439	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	4, 1, 6	grpidpropd 18720	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
8	2, 3, 7	oveq123d 7432	. . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿)))
9	1, 8	mpteq12dv 5202	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))))
10		eqid 2769	. . 3 ⊢ (norm‘𝐾) = (norm‘𝐾)
11		eqid 2769	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2769	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
13		eqid 2769	. . 3 ⊢ (dist‘𝐾) = (dist‘𝐾)
14	10, 11, 12, 13	nmfval 24714	. 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾)))
15		eqid 2769	. . 3 ⊢ (norm‘𝐿) = (norm‘𝐿)
16		eqid 2769	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
17		eqid 2769	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
18		eqid 2769	. . 3 ⊢ (dist‘𝐿) = (dist‘𝐿)
19	15, 16, 17, 18	nmfval 24714	. 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))
20	9, 14, 19	3eqtr4g 2829	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  distcds 17319  0_gc0g 17492  normcnm 24702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-0g 17494  df-nm 24708

This theorem is referenced by:  sranlm  24810  rlmnm  24815  zlmnm  34299