Theorem nrgabv 24178

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 24177	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
4	3	simprbi 498	1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ‘cfv 6544  AbsValcabv 20424  normcnm 24085  NrmGrpcngp 24086  NrmRingcnrg 24088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-nrg 24094

This theorem is referenced by:  nrgring  24180  nmmul  24181  nm1  24184  nrgdomn  24188  subrgnrg  24190  sranlm  24201