Theorem isnzr 14139

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 5626	. . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
2		isnzr.o	. . . 4 ⊢ 1 = (1r‘𝑅)
3	1, 2	eqtr4di 2280	. . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
4		fveq2 5626	. . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isnzr.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	4, 5	eqtr4di 2280	. . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	3, 6	neeq12d 2420	. 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 ))
8		df-nzr 14138	. 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
9	7, 8	elrab2 2962	1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ‘cfv 5317  0_gc0g 13284  1_rcur 13917  Ringcrg 13954  NzRingcnzr 14137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-nzr 14138

This theorem is referenced by:  nzrnz  14140  isnzr2  14142  opprnzrbg  14143  ringelnzr  14145  subrgnzr  14200  zringnzr  14560