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Theorem isnzr 19191
 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 6153 . . . 4 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
2 isnzr.o . . . 4 1 = (1r𝑅)
31, 2syl6eqr 2673 . . 3 (𝑟 = 𝑅 → (1r𝑟) = 1 )
4 fveq2 6153 . . . 4 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isnzr.z . . . 4 0 = (0g𝑅)
64, 5syl6eqr 2673 . . 3 (𝑟 = 𝑅 → (0g𝑟) = 0 )
73, 6neeq12d 2851 . 2 (𝑟 = 𝑅 → ((1r𝑟) ≠ (0g𝑟) ↔ 10 ))
8 df-nzr 19190 . 2 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
97, 8elrab2 3352 1 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ‘cfv 5852  0gc0g 16032  1rcur 18433  Ringcrg 18479  NzRingcnzr 19189 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-nzr 19190 This theorem is referenced by:  nzrnz  19192  nzrring  19193  drngnzr  19194  isnzr2  19195  isnzr2hash  19196  ringelnzr  19198  subrgnzr  19200  zringnzr  19762  chrnzr  19810  nrginvrcn  22419  ply1nzb  23803  zrhnm  29819  isdomn3  37298
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