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 19511 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
3 | zrhker.1 | . . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
4 | 3 | zrhrhm 20661 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) |
5 | zringbas 20625 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
6 | zrhker.0 | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | 5, 6 | rhmf 19480 | . . . 4 ⊢ (𝐿 ∈ (ℤring RingHom 𝑅) → 𝐿:ℤ⟶𝐵) |
8 | ffn 6516 | . . . 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 4583 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ {0} ↔ 𝑀 = 0)) | |
13 | 12 | necon3bbid 3055 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (¬ 𝑀 ∈ {0} ↔ 𝑀 ≠ 0)) |
14 | 13 | biimpar 480 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ¬ 𝑀 ∈ {0}) |
15 | 14 | adantl 484 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ¬ 𝑀 ∈ {0}) |
16 | 11, 15 | eldifd 3949 | . . 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 31221 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) |
19 | 18 | adantr 483 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) |
20 | 16, 19 | eleqtrrd 2918 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → 𝑀 ∈ (◡𝐿 “ (Unit‘𝑅))) |
21 | elpreima 6830 | . . 3 ⊢ (𝐿 Fn ℤ → (𝑀 ∈ (◡𝐿 “ (Unit‘𝑅)) ↔ (𝑀 ∈ ℤ ∧ (𝐿‘𝑀) ∈ (Unit‘𝑅)))) | |
22 | 21 | simplbda 502 | . 2 ⊢ ((𝐿 Fn ℤ ∧ 𝑀 ∈ (◡𝐿 “ (Unit‘𝑅))) → (𝐿‘𝑀) ∈ (Unit‘𝑅)) |
23 | 10, 20, 22 | syl2anc 586 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ◡ccnv 5556 “ cima 5560 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ℤcz 11984 Basecbs 16485 0gc0g 16715 Ringcrg 19299 Unitcui 19391 RingHom crh 19466 DivRingcdr 19504 ℤringzring 20619 ℤRHomczrh 20649 chrcchr 20651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-od 18658 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-rnghom 19469 df-drng 19506 df-subrg 19535 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-chr 20655 |
This theorem is referenced by: qqhghm 31231 qqhrhm 31232 qqhnm 31233 |
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