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Theorem isnzr 20515
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 6905 . . . 4 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
2 isnzr.o . . . 4 1 = (1r𝑅)
31, 2eqtr4di 2794 . . 3 (𝑟 = 𝑅 → (1r𝑟) = 1 )
4 fveq2 6905 . . . 4 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isnzr.z . . . 4 0 = (0g𝑅)
64, 5eqtr4di 2794 . . 3 (𝑟 = 𝑅 → (0g𝑟) = 0 )
73, 6neeq12d 3001 . 2 (𝑟 = 𝑅 → ((1r𝑟) ≠ (0g𝑟) ↔ 10 ))
8 df-nzr 20514 . 2 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
97, 8elrab2 3694 1 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  cfv 6560  0gc0g 17485  1rcur 20179  Ringcrg 20231  NzRingcnzr 20513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-nzr 20514
This theorem is referenced by:  nzrnz  20516  nzrringOLD  20518  isnzr2  20519  isnzr2hash  20520  nzrpropd  20521  opprnzrb  20522  ringelnzr  20524  subrgnzr  20595  isdomn3  20716  drngnzr  20749  zringnzr  21472  chrnzr  21546  nrginvrcn  24714  ply1nzb  26163  drngidlhash  33463  qsnzr  33484  mxidlnzr  33496  zrhnm  33969
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