Theorem nrgabv 24829

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 24828	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
4	3	simprbi 502	1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ‘cfv 6536  AbsValcabv 20922  normcnm 24744  NrmGrpcngp 24745  NrmRingcnrg 24747

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-nrg 24753

This theorem is used by:  nrgring  24831  nmmul  24832  nm1  24835  nrgdomn  24839  subrgnrg  24841  sranlm  24852