Theorem nrgabv 24891

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 24890	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
4	3	simprbi 503	1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  AbsValcabv 20978  normcnm 24806  NrmGrpcngp 24807  NrmRingcnrg 24809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-nrg 24815

This theorem is used by:  nrgring  24893  nmmul  24894  nm1  24897  nrgdomn  24901  subrgnrg  24903  sranlm  24914