Theorem isnrg 23269

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 6670	. . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2		isnrg.1	. . . 4 ⊢ 𝑁 = (norm‘𝑅)
3	1, 2	syl6eqr 2874	. . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4		fveq2 6670	. . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5		isnrg.2	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
6	4, 5	syl6eqr 2874	. . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
7	3, 6	eleq12d 2907	. 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴))
8		df-nrg 23195	. 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
9	7, 8	elrab2 3683	1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ‘cfv 6355  AbsValcabv 19587  normcnm 23186  NrmGrpcngp 23187  NrmRingcnrg 23189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-nrg 23195

This theorem is referenced by:  nrgabv  23270  nrgngp  23271  subrgnrg  23282  tngnrg  23283  cnnrg  23389  zhmnrg  31208