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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 2772 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | nzrunit.2 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | nzrnz 19744 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
4 | nzrring 19745 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
5 | nzrunit.1 | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
6 | 5, 2, 1 | 0unit 19143 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ (1r‘𝑅) = 0 )) |
7 | 6 | necon3bbid 2998 | . . . . . 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 249 | . . . 4 ⊢ (𝑅 ∈ NzRing → ¬ 0 ∈ 𝑈) |
10 | eleq1 2847 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝑈 ↔ 0 ∈ 𝑈)) | |
11 | 10 | notbid 310 | . . . 4 ⊢ (𝐴 = 0 → (¬ 𝐴 ∈ 𝑈 ↔ ¬ 0 ∈ 𝑈)) |
12 | 9, 11 | syl5ibrcom 239 | . . 3 ⊢ (𝑅 ∈ NzRing → (𝐴 = 0 → ¬ 𝐴 ∈ 𝑈)) |
13 | 12 | necon2ad 2976 | . 2 ⊢ (𝑅 ∈ NzRing → (𝐴 ∈ 𝑈 → 𝐴 ≠ 0 )) |
14 | 13 | imp 398 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ‘cfv 6182 0gc0g 16559 1rcur 18964 Ringcrg 19010 Unitcui 19102 NzRingcnzr 19741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-minusg 17885 df-mgp 18953 df-ur 18965 df-ring 19012 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-nzr 19742 |
This theorem is referenced by: unitnmn0 22970 nrginvrcnlem 22993 nzrneg1ne0 43444 |
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