Theorem nzrringOLD 20518

Description: Obsolete version of nzrring 20517 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 2736	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2736	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3	1, 2	isnzr 20515	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2107   ≠ wne 2939  ‘cfv 6560  0_gc0g 17485  1_rcur 20179  Ringcrg 20231  NzRingcnzr 20513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-nzr 20514

This theorem is referenced by: (None)