Theorem isnrg 24168

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 6888	. . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2		isnrg.1	. . . 4 ⊢ 𝑁 = (norm‘𝑅)
3	1, 2	eqtr4di 2790	. . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4		fveq2 6888	. . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5		isnrg.2	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
6	4, 5	eqtr4di 2790	. . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
7	3, 6	eleq12d 2827	. 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴))
8		df-nrg 24085	. 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
9	7, 8	elrab2 3685	1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  AbsValcabv 20416  normcnm 24076  NrmGrpcngp 24077  NrmRingcnrg 24079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-nrg 24085

This theorem is referenced by:  nrgabv  24169  nrgngp  24170  subrgnrg  24181  tngnrg  24182  cnnrg  24288  zhmnrg  32935