Theorem nrgabv 22275

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 22274	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
4	3	simprbi 479	1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  AbsValcabv 18639  normcnm 22191  NrmGrpcngp 22192  NrmRingcnrg 22194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-nrg 22200

This theorem is referenced by:  nrgring  22277  nmmul  22278  nm1  22281  nrgdomn  22285  subrgnrg  22287  sranlm  22298