Theorem isnzr 20460

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 6902	. . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
2		isnzr.o	. . . 4 ⊢ 1 = (1r‘𝑅)
3	1, 2	eqtr4di 2786	. . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
4		fveq2 6902	. . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isnzr.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	4, 5	eqtr4di 2786	. . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	3, 6	neeq12d 2999	. 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 ))
8		df-nzr 20459	. 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
9	7, 8	elrab2 3687	1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ‘cfv 6553  0_gc0g 17428  1_rcur 20128  Ringcrg 20180  NzRingcnzr 20458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-nzr 20459

This theorem is referenced by:  nzrnz  20461  nzrringOLD  20463  isnzr2  20464  isnzr2hash  20465  ringelnzr  20467  subrgnzr  20540  drngnzr  20651  isdomn3  21255  zringnzr  21393  chrnzr  21467  nrginvrcn  24629  ply1nzb  26078  drngidlhash  33175  qsnzr  33196  mxidlnzr  33205  zrhnm  33603