Theorem nzrringOLD 20728

Description: Obsolete version of nzrring 20727 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 20725	. 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 2145   ≠ wne 2955  ‘cfv 6527  0_gc0g 17571  1_rcur 20368  Ringcrg 20420  NzRingcnzr 20723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-nzr 20724

This theorem is used by: (None)