Theorem isnzr 19080

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 6103	. . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
2		isnzr.o	. . . 4 ⊢ 1 = (1r‘𝑅)
3	1, 2	syl6eqr 2662	. . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
4		fveq2 6103	. . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isnzr.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	4, 5	syl6eqr 2662	. . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	3, 6	neeq12d 2843	. 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 ))
8		df-nzr 19079	. 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
9	7, 8	elrab2 3333	1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ‘cfv 5804  0_gc0g 15923  1_rcur 18324  Ringcrg 18370  NzRingcnzr 19078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-nzr 19079

This theorem is referenced by:  nzrnz  19081  nzrring  19082  drngnzr  19083  isnzr2  19084  isnzr2hash  19085  ringelnzr  19087  subrgnzr  19089  zringnzr  19649  chrnzr  19697  nrginvrcn  22306  ply1nzb  23686  zrhnm  29341  isdomn3  36801