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| 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.) | 
| Ref | Expression | 
|---|---|
| nzrringOLD | ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | 
| 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2939 ‘cfv 6560 0gc0g 17485 1rcur 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) | 
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