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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 765 | . . 3 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β π β DivRing) | |
2 | drngring 20368 | . . 3 β’ (π β DivRing β π β Ring) | |
3 | zrhker.1 | . . . . 5 β’ πΏ = (β€RHomβπ ) | |
4 | 3 | zrhrhm 21067 | . . . 4 β’ (π β Ring β πΏ β (β€ring RingHom π )) |
5 | zringbas 21029 | . . . . 5 β’ β€ = (Baseββ€ring) | |
6 | zrhker.0 | . . . . 5 β’ π΅ = (Baseβπ ) | |
7 | 5, 6 | rhmf 20267 | . . . 4 β’ (πΏ β (β€ring RingHom π ) β πΏ:β€βΆπ΅) |
8 | ffn 6717 | . . . 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 769 | . . . 4 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ (π β β€ β§ π β 0)) β π β β€) | |
12 | elsng 4642 | . . . . . . 7 β’ (π β β€ β (π β {0} β π = 0)) | |
13 | 12 | necon3bbid 2978 | . . . . . 6 β’ (π β β€ β (Β¬ π β {0} β π β 0)) |
14 | 13 | biimpar 478 | . . . . 5 β’ ((π β β€ β§ π β 0) β Β¬ π β {0}) |
15 | 14 | adantl 482 | . . . 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 3959 | . . 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 33027 | . . . 4 β’ ((π β DivRing β§ (chrβπ ) = 0) β (β‘πΏ β (Unitβπ )) = (β€ β {0})) |
19 | 18 | adantr 481 | . . 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 2836 | . 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 7059 | . . 3 β’ (πΏ Fn β€ β (π β (β‘πΏ β (Unitβπ )) β (π β β€ β§ (πΏβπ) β (Unitβπ )))) | |
22 | 21 | simplbda 500 | . 2 β’ ((πΏ Fn β€ β§ π β (β‘πΏ β (Unitβπ ))) β (πΏβπ) β (Unitβπ )) |
23 | 10, 20, 22 | syl2anc 584 | 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 396 = wceq 1541 β wcel 2106 β wne 2940 β cdif 3945 {csn 4628 β‘ccnv 5675 β cima 5679 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7411 0cc0 11112 β€cz 12560 Basecbs 17146 0gc0g 17387 Ringcrg 20058 Unitcui 20173 RingHom crh 20252 DivRingcdr 20361 β€ringczring 21023 β€RHomczrh 21055 chrcchr 21057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-fz 13487 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16200 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-ghm 19092 df-od 19398 df-cmn 19652 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-rnghom 20255 df-subrg 20321 df-drng 20363 df-cnfld 20951 df-zring 21024 df-zrh 21059 df-chr 21061 |
This theorem is referenced by: qqhghm 33037 qqhrhm 33038 qqhnm 33039 |
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