Theorem isnrg 24782

Description: A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
isnrg	⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Proof of Theorem isnrg

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6879	. . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2		isnrg.1	. . . 4 ⊢ 𝑁 = (norm‘𝑅)
3	1, 2	eqtr4di 2822	. . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4		fveq2 6879	. . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5		isnrg.2	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
6	4, 5	eqtr4di 2822	. . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
7	3, 6	eleq12d 2863	. 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴))
8		df-nrg 24707	. 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
9	7, 8	elrab2 3663	1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ‘cfv 6534  AbsValcabv 20885  normcnm 24698  NrmGrpcngp 24699  NrmRingcnrg 24701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-nrg 24707

This theorem is referenced by:  nrgabv  24783  nrgngp  24784  subrgnrg  24795  tngnrg  24796  cnnrg  24902  zhmnrg  34296