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| Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| isnzr.o | ⊢ 1 = (1r‘𝑅) | 
| isnzr.z | ⊢ 0 = (0g‘𝑅) | 
| Ref | Expression | 
|---|---|
| isnzr | ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
| 2 | isnzr.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
| 3 | 1, 2 | eqtr4di 2794 | . . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) | 
| 4 | fveq2 6905 | . . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
| 5 | isnzr.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
| 6 | 4, 5 | eqtr4di 2794 | . . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) | 
| 7 | 3, 6 | neeq12d 3001 | . 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 )) | 
| 8 | df-nzr 20514 | . 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} | |
| 9 | 7, 8 | elrab2 3694 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ 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: nzrnz 20516 nzrringOLD 20518 isnzr2 20519 isnzr2hash 20520 nzrpropd 20521 opprnzrb 20522 ringelnzr 20524 subrgnzr 20595 isdomn3 20716 drngnzr 20749 zringnzr 21472 chrnzr 21546 nrginvrcn 24714 ply1nzb 26163 drngidlhash 33463 qsnzr 33484 mxidlnzr 33496 zrhnm 33969 | 
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