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| Mirrors > Home > MPE Home > Th. List > nzrnz | Structured version Visualization version GIF version | ||
| Description: One and zero are different in 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 |
|---|---|
| nzrnz | ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isnzr.o | . . 3 ⊢ 1 = (1r‘𝑅) | |
| 2 | isnzr.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
| 3 | 1, 2 | isnzr 20514 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) |
| 4 | 3 | simprbi 496 | 1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 0gc0g 17484 1rcur 20178 Ringcrg 20230 NzRingcnzr 20512 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-nzr 20513 |
| This theorem is referenced by: nzrunit 20524 nrhmzr 20537 lringnz 20543 subrgnzr 20594 rrgnz 20704 fidomndrng 20774 uvcf1 21812 lindfind2 21838 nm1 24688 deg1pw 26160 ply1nz 26161 ply1nzb 26162 mon1pid 26193 lgsqrlem4 27393 unitnz 33243 domnprodn0 33279 fracfld 33310 drngidl 33461 drngidlhash 33462 drnglidl1ne0 33503 drng0mxidl 33504 qsdrngi 33523 ply1moneq 33611 deg1vr 33614 dimlssid 33683 ply1annnr 33746 algextdeglem4 33761 rtelextdg2lem 33767 zrhnm 33968 idomnnzpownz 42133 idomnnzgmulnz 42134 deg1gprod 42141 deg1pow 42142 domnexpgn0cl 42533 abvexp 42542 fiabv 42546 uvcn0 42552 deg1mhm 43212 |
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