Theorem nmpropd 24751

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 2764	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
4		eqidd 2764	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5		nmpropd.2	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))
6	5	oveqdr 7438	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	4, 1, 6	grpidpropd 18715	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
8	2, 3, 7	oveq123d 7431	. . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿)))
9	1, 8	mpteq12dv 5198	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))))
10		eqid 2763	. . 3 ⊢ (norm‘𝐾) = (norm‘𝐾)
11		eqid 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2763	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
13		eqid 2763	. . 3 ⊢ (dist‘𝐾) = (dist‘𝐾)
14	10, 11, 12, 13	nmfval 24745	. 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾)))
15		eqid 2763	. . 3 ⊢ (norm‘𝐿) = (norm‘𝐿)
16		eqid 2763	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
17		eqid 2763	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
18		eqid 2763	. . 3 ⊢ (dist‘𝐿) = (dist‘𝐿)
19	15, 16, 17, 18	nmfval 24745	. 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))
20	9, 14, 19	3eqtr4g 2823	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  +_gcplusg 17305  distcds 17314  0_gc0g 17487  normcnm 24733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-0g 17489  df-nm 24739

This theorem is referenced by:  sranlm  24841  rlmnm  24846  zlmnm  34354