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Theorem isnzr 19461
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 6352 . . . 4 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
2 isnzr.o . . . 4 1 = (1r𝑅)
31, 2syl6eqr 2812 . . 3 (𝑟 = 𝑅 → (1r𝑟) = 1 )
4 fveq2 6352 . . . 4 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isnzr.z . . . 4 0 = (0g𝑅)
64, 5syl6eqr 2812 . . 3 (𝑟 = 𝑅 → (0g𝑟) = 0 )
73, 6neeq12d 2993 . 2 (𝑟 = 𝑅 → ((1r𝑟) ≠ (0g𝑟) ↔ 10 ))
8 df-nzr 19460 . 2 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
97, 8elrab2 3507 1 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  cfv 6049  0gc0g 16302  1rcur 18701  Ringcrg 18747  NzRingcnzr 19459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-nzr 19460
This theorem is referenced by:  nzrnz  19462  nzrring  19463  drngnzr  19464  isnzr2  19465  isnzr2hash  19466  ringelnzr  19468  subrgnzr  19470  zringnzr  20032  chrnzr  20080  nrginvrcn  22697  ply1nzb  24081  zrhnm  30322  isdomn3  38284
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