Theorem nzrringOLD 20434

Description: Obsolete version of nzrring 20433 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 2733	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2733	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3	1, 2	isnzr 20431	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 2929  ‘cfv 6486  0_gc0g 17345  1_rcur 20101  Ringcrg 20153  NzRingcnzr 20429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-nzr 20430

This theorem is referenced by: (None)