Theorem nzrringOLD 20454

Description: Obsolete version of nzrring 20453 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 2737	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2737	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3	1, 2	isnzr 20451	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6493  0_gc0g 17363  1_rcur 20120  Ringcrg 20172  NzRingcnzr 20449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-nzr 20450

This theorem is referenced by: (None)