Theorem nzrringOLD 20767

Description: Obsolete version of nzrring 20766 as of 23-Feb-2025. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nzrringOLD	⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Proof of Theorem nzrringOLD

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2760	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3	1, 2	isnzr 20764	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)))
4	3	simplbi 502	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955  ‘cfv 6538  0_gc0g 17610  1_rcur 20407  Ringcrg 20459  NzRingcnzr 20762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-nzr 20763

This theorem is used by: (None)