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Mirrors > Home > ILE Home > Th. List > nzrnz | 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 13661 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) |
4 | 3 | simprbi 275 | 1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 ‘cfv 5246 0gc0g 12857 1rcur 13439 Ringcrg 13476 NzRingcnzr 13659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-rex 2478 df-rab 2481 df-v 2762 df-un 3157 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-iota 5207 df-fv 5254 df-nzr 13660 |
This theorem is referenced by: nzrunit 13668 lringnz 13675 subrgnzr 13722 rrgnz 13748 |
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