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Theorem isnzr 20652
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 6838 . . . 4 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
2 isnzr.o . . . 4 1 = (1r𝑅)
31, 2eqtr4di 2796 . . 3 (𝑟 = 𝑅 → (1r𝑟) = 1 )
4 fveq2 6838 . . . 4 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isnzr.z . . . 4 0 = (0g𝑅)
64, 5eqtr4di 2796 . . 3 (𝑟 = 𝑅 → (0g𝑟) = 0 )
73, 6neeq12d 3004 . 2 (𝑟 = 𝑅 → ((1r𝑟) ≠ (0g𝑟) ↔ 10 ))
8 df-nzr 20651 . 2 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
97, 8elrab2 3647 1 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2942  cfv 6492  0gc0g 17256  1rcur 19842  Ringcrg 19888  NzRingcnzr 20650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6444  df-fv 6500  df-nzr 20651
This theorem is referenced by:  nzrnz  20653  nzrring  20654  drngnzr  20655  isnzr2  20656  isnzr2hash  20657  ringelnzr  20659  subrgnzr  20661  zringnzr  20804  chrnzr  20856  nrginvrcn  23978  ply1nzb  25409  mxidlnzr  32013  zrhnm  32311  isdomn3  41365
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