Theorem nrgabv 22842

Description: The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
isnrg.1	⊢ 𝑁 = (norm‘𝑅)
isnrg.2	⊢ 𝐴 = (AbsVal‘𝑅)

Assertion

Ref	Expression
nrgabv	⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴)

Proof of Theorem nrgabv

Step	Hyp	Ref	Expression
1		isnrg.1	. . 3 ⊢ 𝑁 = (norm‘𝑅)
2		isnrg.2	. . 3 ⊢ 𝐴 = (AbsVal‘𝑅)
3	1, 2	isnrg 22841	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
4	3	simprbi 492	1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ‘cfv 6127  AbsValcabv 19179  normcnm 22758  NrmGrpcngp 22759  NrmRingcnrg 22761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-nrg 22767

This theorem is referenced by:  nrgring  22844  nmmul  22845  nm1  22848  nrgdomn  22852  subrgnrg  22854  sranlm  22865