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Theorem isnrg 23264
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 ∧ 𝑁𝐴))

Proof of Theorem isnrg
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6652 . . . 4 (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2 isnrg.1 . . . 4 𝑁 = (norm‘𝑅)
31, 2eqtr4di 2875 . . 3 (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4 fveq2 6652 . . . 4 (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5 isnrg.2 . . . 4 𝐴 = (AbsVal‘𝑅)
64, 5eqtr4di 2875 . . 3 (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
73, 6eleq12d 2908 . 2 (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁𝐴))
8 df-nrg 23190 . 2 NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
97, 8elrab2 3658 1 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2114  cfv 6334  AbsValcabv 19578  normcnm 23181  NrmGrpcngp 23182  NrmRingcnrg 23184
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-nrg 23190
This theorem is referenced by:  nrgabv  23265  nrgngp  23266  subrgnrg  23277  tngnrg  23278  cnnrg  23384  zhmnrg  31282
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