Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nzrunit | Structured version Visualization version GIF version |
Description: A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
nzrunit.1 | ⊢ 𝑈 = (Unit‘𝑅) |
nzrunit.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
nzrunit | ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | nzrunit.2 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | nzrnz 20025 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
4 | nzrring 20026 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
5 | nzrunit.1 | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
6 | 5, 2, 1 | 0unit 19422 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ (1r‘𝑅) = 0 )) |
7 | 6 | necon3bbid 3051 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (¬ 0 ∈ 𝑈 ↔ (1r‘𝑅) ≠ 0 )) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (¬ 0 ∈ 𝑈 ↔ (1r‘𝑅) ≠ 0 )) |
9 | 3, 8 | mpbird 259 | . . . 4 ⊢ (𝑅 ∈ NzRing → ¬ 0 ∈ 𝑈) |
10 | eleq1 2898 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝑈 ↔ 0 ∈ 𝑈)) | |
11 | 10 | notbid 320 | . . . 4 ⊢ (𝐴 = 0 → (¬ 𝐴 ∈ 𝑈 ↔ ¬ 0 ∈ 𝑈)) |
12 | 9, 11 | syl5ibrcom 249 | . . 3 ⊢ (𝑅 ∈ NzRing → (𝐴 = 0 → ¬ 𝐴 ∈ 𝑈)) |
13 | 12 | necon2ad 3029 | . 2 ⊢ (𝑅 ∈ NzRing → (𝐴 ∈ 𝑈 → 𝐴 ≠ 0 )) |
14 | 13 | imp 409 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ‘cfv 6348 0gc0g 16705 1rcur 19243 Ringcrg 19289 Unitcui 19381 NzRingcnzr 20022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-nzr 20023 |
This theorem is referenced by: unitnmn0 23269 nrginvrcnlem 23292 nzrneg1ne0 44131 |
Copyright terms: Public domain | W3C validator |