Theorem isnzr 20034

Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
isnzr.o	⊢ 1 = (1r‘𝑅)
isnzr.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
isnzr	⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))

Proof of Theorem isnzr

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6672	. . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
2		isnzr.o	. . . 4 ⊢ 1 = (1r‘𝑅)
3	1, 2	syl6eqr 2876	. . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
4		fveq2 6672	. . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isnzr.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	4, 5	syl6eqr 2876	. . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	3, 6	neeq12d 3079	. 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 ))
8		df-nzr 20033	. 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
9	7, 8	elrab2 3685	1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ‘cfv 6357  0_gc0g 16715  1_rcur 19253  Ringcrg 19299  NzRingcnzr 20032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-nzr 20033

This theorem is referenced by:  nzrnz  20035  nzrring  20036  drngnzr  20037  isnzr2  20038  isnzr2hash  20039  ringelnzr  20041  subrgnzr  20043  zringnzr  20631  chrnzr  20679  nrginvrcn  23303  ply1nzb  24718  mxidlnzr  30978  zrhnm  31212  isdomn3  39811