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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 β€RHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
Ref | Expression |
---|---|
zrhker.0 | β’ π΅ = (Baseβπ ) |
zrhker.1 | β’ πΏ = (β€RHomβπ ) |
zrhker.2 | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
elzrhunit | β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β (πΏβπ) β (Unitβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . . 3 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β π β DivRing) | |
2 | drngring 20206 | . . 3 β’ (π β DivRing β π β Ring) | |
3 | zrhker.1 | . . . . 5 β’ πΏ = (β€RHomβπ ) | |
4 | 3 | zrhrhm 20928 | . . . 4 β’ (π β Ring β πΏ β (β€ring RingHom π )) |
5 | zringbas 20891 | . . . . 5 β’ β€ = (Baseββ€ring) | |
6 | zrhker.0 | . . . . 5 β’ π΅ = (Baseβπ ) | |
7 | 5, 6 | rhmf 20167 | . . . 4 β’ (πΏ β (β€ring RingHom π ) β πΏ:β€βΆπ΅) |
8 | ffn 6673 | . . . 4 β’ (πΏ:β€βΆπ΅ β πΏ Fn β€) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 β’ (π β Ring β πΏ Fn β€) |
10 | 1, 2, 9 | 3syl 18 | . 2 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β πΏ Fn β€) |
11 | simprl 770 | . . . 4 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β π β β€) | |
12 | elsng 4605 | . . . . . . 7 β’ (π β β€ β (π β {0} β π = 0)) | |
13 | 12 | necon3bbid 2982 | . . . . . 6 β’ (π β β€ β (Β¬ π β {0} β π β 0)) |
14 | 13 | biimpar 479 | . . . . 5 β’ ((π β β€ β§ π β 0) β Β¬ π β {0}) |
15 | 14 | adantl 483 | . . . 4 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β Β¬ π β {0}) |
16 | 11, 15 | eldifd 3926 | . . 3 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β π β (β€ β {0})) |
17 | zrhker.2 | . . . . 5 β’ 0 = (0gβπ ) | |
18 | 6, 3, 17 | zrhunitpreima 32599 | . . . 4 β’ ((π β DivRing β§ (chrβπ ) = 0) β (β‘πΏ β (Unitβπ )) = (β€ β {0})) |
19 | 18 | adantr 482 | . . 3 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β (β‘πΏ β (Unitβπ )) = (β€ β {0})) |
20 | 16, 19 | eleqtrrd 2841 | . 2 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β π β (β‘πΏ β (Unitβπ ))) |
21 | elpreima 7013 | . . 3 β’ (πΏ Fn β€ β (π β (β‘πΏ β (Unitβπ )) β (π β β€ β§ (πΏβπ) β (Unitβπ )))) | |
22 | 21 | simplbda 501 | . 2 β’ ((πΏ Fn β€ β§ π β (β‘πΏ β (Unitβπ ))) β (πΏβπ) β (Unitβπ )) |
23 | 10, 20, 22 | syl2anc 585 | 1 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β (πΏβπ) β (Unitβπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 β cdif 3912 {csn 4591 β‘ccnv 5637 β cima 5641 Fn wfn 6496 βΆwf 6497 βcfv 6501 (class class class)co 7362 0cc0 11058 β€cz 12506 Basecbs 17090 0gc0g 17328 Ringcrg 19971 Unitcui 20075 RingHom crh 20152 DivRingcdr 20199 β€ringczring 20885 β€RHomczrh 20916 chrcchr 20918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-fz 13432 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-ghm 19013 df-od 19317 df-cmn 19571 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-rnghom 20155 df-drng 20201 df-subrg 20236 df-cnfld 20813 df-zring 20886 df-zrh 20920 df-chr 20922 |
This theorem is referenced by: qqhghm 32609 qqhrhm 32610 qqhnm 32611 |
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