![]() |
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 2778 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2778 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 19656 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | 3 | simplbi 493 | 1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6135 0gc0g 16486 1rcur 18888 Ringcrg 18934 NzRingcnzr 19654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-nzr 19655 |
This theorem is referenced by: opprnzr 19662 nzrunit 19664 domnring 19693 domnchr 20276 uvcf1 20535 lindfind2 20561 frlmisfrlm 20591 nminvr 22881 deg1pw 24317 ply1nz 24318 ply1remlem 24359 ply1rem 24360 facth1 24361 fta1glem1 24362 fta1glem2 24363 zrhnm 30611 mon1pid 38742 mon1psubm 38743 nzrneg1ne0 42884 islindeps2 43287 |
Copyright terms: Public domain | W3C validator |