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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elzrhunit | Structured version Visualization version GIF version |
Description: Condition for the image by ℤRHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
Ref | Expression |
---|---|
zrhker.0 | ⊢ 𝐵 = (Base‘𝑅) |
zrhker.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
zrhker.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
elzrhunit | ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝐿‘𝑀) ∈ (Unit‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → 𝑅 ∈ DivRing) | |
2 | drngring 20643 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
3 | zrhker.1 | . . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
4 | 3 | zrhrhm 21454 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
5 | zringbas 21396 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
6 | zrhker.0 | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | 5, 6 | rhmf 20436 | . . . 4 ⊢ (𝐿 ∈ (ℤring RingHom 𝑅) → 𝐿:ℤ⟶𝐵) |
8 | ffn 6723 | . . . 4 ⊢ (𝐿:ℤ⟶𝐵 → 𝐿 Fn ℤ) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐿 Fn ℤ) |
10 | 1, 2, 9 | 3syl 18 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → 𝐿 Fn ℤ) |
11 | simprl 769 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → 𝑀 ∈ ℤ) | |
12 | elsng 4644 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ {0} ↔ 𝑀 = 0)) | |
13 | 12 | necon3bbid 2967 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (¬ 𝑀 ∈ {0} ↔ 𝑀 ≠ 0)) |
14 | 13 | biimpar 476 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ¬ 𝑀 ∈ {0}) |
15 | 14 | adantl 480 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ¬ 𝑀 ∈ {0}) |
16 | 11, 15 | eldifd 3955 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → 𝑀 ∈ (ℤ ∖ {0})) |
17 | zrhker.2 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
18 | 6, 3, 17 | zrhunitpreima 33710 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) |
19 | 18 | adantr 479 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) |
20 | 16, 19 | eleqtrrd 2828 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → 𝑀 ∈ (◡𝐿 “ (Unit‘𝑅))) |
21 | elpreima 7066 | . . 3 ⊢ (𝐿 Fn ℤ → (𝑀 ∈ (◡𝐿 “ (Unit‘𝑅)) ↔ (𝑀 ∈ ℤ ∧ (𝐿‘𝑀) ∈ (Unit‘𝑅)))) | |
22 | 21 | simplbda 498 | . 2 ⊢ ((𝐿 Fn ℤ ∧ 𝑀 ∈ (◡𝐿 “ (Unit‘𝑅))) → (𝐿‘𝑀) ∈ (Unit‘𝑅)) |
23 | 10, 20, 22 | syl2anc 582 | 1 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝐿‘𝑀) ∈ (Unit‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3941 {csn 4630 ◡ccnv 5677 “ cima 5681 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 0cc0 11140 ℤcz 12591 Basecbs 17183 0gc0g 17424 Ringcrg 20185 Unitcui 20306 RingHom crh 20420 DivRingcdr 20636 ℤringczring 21389 ℤRHomczrh 21442 chrcchr 21444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fz 13520 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-ghm 19176 df-od 19495 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-rhm 20423 df-subrng 20495 df-subrg 20520 df-drng 20638 df-cnfld 21297 df-zring 21390 df-zrh 21446 df-chr 21448 |
This theorem is referenced by: qqhghm 33720 qqhrhm 33721 qqhnm 33722 |
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