Theorem nmpropd 24922

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 2761	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
4		eqidd 2761	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5		nmpropd.2	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))
6	5	oveqdr 7452	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	4, 1, 6	grpidpropd 18851	. . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
8	2, 3, 7	oveq123d 7445	. . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿)))
9	1, 8	mpteq12dv 5191	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))))
10		eqid 2760	. . 3 ⊢ (norm‘𝐾) = (norm‘𝐾)
11		eqid 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2760	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
13		eqid 2760	. . 3 ⊢ (dist‘𝐾) = (dist‘𝐾)
14	10, 11, 12, 13	nmfval 24916	. 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾)))
15		eqid 2760	. . 3 ⊢ (norm‘𝐿) = (norm‘𝐿)
16		eqid 2760	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
17		eqid 2760	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
18		eqid 2760	. . 3 ⊢ (dist‘𝐿) = (dist‘𝐿)
19	15, 16, 17, 18	nmfval 24916	. 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))
20	9, 14, 19	3eqtr4g 2820	1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  distcds 17446  0_gc0g 17619  normcnm 24904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7427  df-0g 17621  df-nm 24910

This theorem is used by:  sranlm  25012  rlmnm  25017  zlmnm  34646