Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nzrring | Structured version Visualization version GIF version |
Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
nzrring | ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2818 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 19960 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | 3 | simplbi 498 | 1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 0gc0g 16701 1rcur 19180 Ringcrg 19226 NzRingcnzr 19958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-nzr 19959 |
This theorem is referenced by: opprnzr 19966 nzrunit 19968 domnring 19997 domnchr 20607 uvcf1 20864 lindfind2 20890 frlmisfrlm 20920 nminvr 23205 deg1pw 24641 ply1nz 24642 ply1remlem 24683 ply1rem 24684 facth1 24685 fta1glem1 24686 fta1glem2 24687 zrhnm 31109 uvcn0 39029 0prjspnlem 39146 mon1pid 39683 mon1psubm 39684 nzrneg1ne0 44068 islindeps2 44466 |
Copyright terms: Public domain | W3C validator |