Theorem isnrg 23869

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 6804	. . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2		isnrg.1	. . . 4 ⊢ 𝑁 = (norm‘𝑅)
3	1, 2	eqtr4di 2794	. . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4		fveq2 6804	. . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5		isnrg.2	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
6	4, 5	eqtr4di 2794	. . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
7	3, 6	eleq12d 2831	. 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴))
8		df-nrg 23786	. 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
9	7, 8	elrab2 3632	1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ‘cfv 6458  AbsValcabv 20121  normcnm 23777  NrmGrpcngp 23778  NrmRingcnrg 23780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6410  df-fv 6466  df-nrg 23786

This theorem is referenced by:  nrgabv  23870  nrgngp  23871  subrgnrg  23882  tngnrg  23883  cnnrg  23989  zhmnrg  31962