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Mirrors > Home > MPE Home > Th. List > isnzr | Structured version Visualization version GIF version |
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 6837 | . . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
2 | isnzr.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
4 | fveq2 6837 | . . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
5 | isnzr.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
6 | 4, 5 | eqtr4di 2795 | . . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
7 | 3, 6 | neeq12d 3003 | . 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 )) |
8 | df-nzr 20651 | . 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} | |
9 | 7, 8 | elrab2 3646 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ‘cfv 6491 0gc0g 17255 1rcur 19842 Ringcrg 19888 NzRingcnzr 20650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6443 df-fv 6499 df-nzr 20651 |
This theorem is referenced by: nzrnz 20653 nzrring 20654 drngnzr 20655 isnzr2 20656 isnzr2hash 20657 ringelnzr 20659 subrgnzr 20661 zringnzr 20804 chrnzr 20856 nrginvrcn 23978 ply1nzb 25409 mxidlnzr 32025 zrhnm 32323 isdomn3 41396 |
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