Theorem isnzr 14572

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 5695	. . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
2		isnzr.o	. . . 4 ⊢ 1 = (1r‘𝑅)
3	1, 2	eqtr4di 2289	. . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
4		fveq2 5695	. . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
5		isnzr.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	4, 5	eqtr4di 2289	. . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
7	3, 6	neeq12d 2440	. 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 ))
8		df-nzr 14571	. 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
9	7, 8	elrab2 2985	1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ‘cfv 5377  0_gc0g 13663  1_rcur 14346  Ringcrg 14384  NzRingcnzr 14570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-nzr 14571

This theorem is used by:  nzrnz  14573  isnzr2  14575  opprnzrbg  14576  ringelnzr  14578  subrgnzr  14634  aprnzr  14683  drngnzr  14703  zringnzr  15021