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Theorem isnzr 19547
Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
isnzr.o 1 = (1r𝑅)
isnzr.z 0 = (0g𝑅)
Assertion
Ref Expression
isnzr (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))

Proof of Theorem isnzr
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6379 . . . 4 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
2 isnzr.o . . . 4 1 = (1r𝑅)
31, 2syl6eqr 2817 . . 3 (𝑟 = 𝑅 → (1r𝑟) = 1 )
4 fveq2 6379 . . . 4 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isnzr.z . . . 4 0 = (0g𝑅)
64, 5syl6eqr 2817 . . 3 (𝑟 = 𝑅 → (0g𝑟) = 0 )
73, 6neeq12d 2998 . 2 (𝑟 = 𝑅 → ((1r𝑟) ≠ (0g𝑟) ↔ 10 ))
8 df-nzr 19546 . 2 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
97, 8elrab2 3525 1 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  cfv 6070  0gc0g 16380  1rcur 18782  Ringcrg 18828  NzRingcnzr 19545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-iota 6033  df-fv 6078  df-nzr 19546
This theorem is referenced by:  nzrnz  19548  nzrring  19549  drngnzr  19550  isnzr2  19551  isnzr2hash  19552  ringelnzr  19554  subrgnzr  19556  zringnzr  20117  chrnzr  20165  nrginvrcn  22789  ply1nzb  24187  zrhnm  30481  isdomn3  38483
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