![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6645 | . . . 4 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
2 | isnzr.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
3 | 1, 2 | eqtr4di 2851 | . . 3 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
4 | fveq2 6645 | . . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
5 | isnzr.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
6 | 4, 5 | eqtr4di 2851 | . . 3 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
7 | 3, 6 | neeq12d 3048 | . 2 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ≠ (0g‘𝑟) ↔ 1 ≠ 0 )) |
8 | df-nzr 20024 | . 2 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} | |
9 | 7, 8 | elrab2 3631 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 0gc0g 16705 1rcur 19244 Ringcrg 19290 NzRingcnzr 20023 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-nzr 20024 |
This theorem is referenced by: nzrnz 20026 nzrring 20027 drngnzr 20028 isnzr2 20029 isnzr2hash 20030 ringelnzr 20032 subrgnzr 20034 zringnzr 20175 chrnzr 20222 nrginvrcn 23298 ply1nzb 24723 mxidlnzr 31047 zrhnm 31320 isdomn3 40148 |
Copyright terms: Public domain | W3C validator |