Theorem isnrg 22872

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 6446	. . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2		isnrg.1	. . . 4 ⊢ 𝑁 = (norm‘𝑅)
3	1, 2	syl6eqr 2832	. . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4		fveq2 6446	. . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5		isnrg.2	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
6	4, 5	syl6eqr 2832	. . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
7	3, 6	eleq12d 2853	. 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴))
8		df-nrg 22798	. 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
9	7, 8	elrab2 3576	1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ‘cfv 6135  AbsValcabv 19208  normcnm 22789  NrmGrpcngp 22790  NrmRingcnrg 22792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-nrg 22798

This theorem is referenced by:  nrgabv  22873  nrgngp  22874  subrgnrg  22885  tngnrg  22886  cnnrg  22992  zhmnrg  30609