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Mirrors > Home > MPE Home > Th. List > zrhpsgnodpm | Structured version Visualization version GIF version |
Description: The sign of an odd permutation embedded into a ring is the additive inverse of the unity element of the ring. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
zrhpsgnevpm.y | β’ π = (β€RHomβπ ) |
zrhpsgnevpm.s | β’ π = (pmSgnβπ) |
zrhpsgnevpm.o | β’ 1 = (1rβπ ) |
zrhpsgnodpm.p | β’ π = (Baseβ(SymGrpβπ)) |
zrhpsgnodpm.i | β’ πΌ = (invgβπ ) |
Ref | Expression |
---|---|
zrhpsgnodpm | β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β ((π β π)βπΉ) = (πΌβ 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . 6 β’ (SymGrpβπ) = (SymGrpβπ) | |
2 | zrhpsgnevpm.s | . . . . . 6 β’ π = (pmSgnβπ) | |
3 | eqid 2733 | . . . . . 6 β’ ((mulGrpββfld) βΎs {1, -1}) = ((mulGrpββfld) βΎs {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 21008 | . . . . 5 β’ (π β Fin β π β ((SymGrpβπ) GrpHom ((mulGrpββfld) βΎs {1, -1}))) |
5 | zrhpsgnodpm.p | . . . . . 6 β’ π = (Baseβ(SymGrpβπ)) | |
6 | eqid 2733 | . . . . . 6 β’ (Baseβ((mulGrpββfld) βΎs {1, -1})) = (Baseβ((mulGrpββfld) βΎs {1, -1})) | |
7 | 5, 6 | ghmf 19020 | . . . . 5 β’ (π β ((SymGrpβπ) GrpHom ((mulGrpββfld) βΎs {1, -1})) β π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1}))) |
8 | 4, 7 | syl 17 | . . . 4 β’ (π β Fin β π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1}))) |
9 | 8 | 3ad2ant2 1135 | . . 3 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1}))) |
10 | eldifi 4090 | . . . 4 β’ (πΉ β (π β (pmEvenβπ)) β πΉ β π) | |
11 | 10 | 3ad2ant3 1136 | . . 3 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β πΉ β π) |
12 | fvco3 6944 | . . 3 β’ ((π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1})) β§ πΉ β π) β ((π β π)βπΉ) = (πβ(πβπΉ))) | |
13 | 9, 11, 12 | syl2anc 585 | . 2 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β ((π β π)βπΉ) = (πβ(πβπΉ))) |
14 | 1, 5, 2 | psgnodpm 21015 | . . . 4 β’ ((π β Fin β§ πΉ β (π β (pmEvenβπ))) β (πβπΉ) = -1) |
15 | 14 | 3adant1 1131 | . . 3 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β (πβπΉ) = -1) |
16 | 15 | fveq2d 6850 | . 2 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β (πβ(πβπΉ)) = (πβ-1)) |
17 | zrhpsgnevpm.y | . . . . . . 7 β’ π = (β€RHomβπ ) | |
18 | 17 | zrhrhm 20935 | . . . . . 6 β’ (π β Ring β π β (β€ring RingHom π )) |
19 | rhmghm 20167 | . . . . . 6 β’ (π β (β€ring RingHom π ) β π β (β€ring GrpHom π )) | |
20 | 18, 19 | syl 17 | . . . . 5 β’ (π β Ring β π β (β€ring GrpHom π )) |
21 | 1z 12541 | . . . . . 6 β’ 1 β β€ | |
22 | 21 | a1i 11 | . . . . 5 β’ (π β Ring β 1 β β€) |
23 | zringbas 20898 | . . . . . 6 β’ β€ = (Baseββ€ring) | |
24 | eqid 2733 | . . . . . 6 β’ (invgββ€ring) = (invgββ€ring) | |
25 | zrhpsgnodpm.i | . . . . . 6 β’ πΌ = (invgβπ ) | |
26 | 23, 24, 25 | ghminv 19023 | . . . . 5 β’ ((π β (β€ring GrpHom π ) β§ 1 β β€) β (πβ((invgββ€ring)β1)) = (πΌβ(πβ1))) |
27 | 20, 22, 26 | syl2anc 585 | . . . 4 β’ (π β Ring β (πβ((invgββ€ring)β1)) = (πΌβ(πβ1))) |
28 | zringinvg 20909 | . . . . . . . 8 β’ (1 β β€ β -1 = ((invgββ€ring)β1)) | |
29 | 21, 28 | ax-mp 5 | . . . . . . 7 β’ -1 = ((invgββ€ring)β1) |
30 | 29 | eqcomi 2742 | . . . . . 6 β’ ((invgββ€ring)β1) = -1 |
31 | 30 | fveq2i 6849 | . . . . 5 β’ (πβ((invgββ€ring)β1)) = (πβ-1) |
32 | 31 | a1i 11 | . . . 4 β’ (π β Ring β (πβ((invgββ€ring)β1)) = (πβ-1)) |
33 | zrhpsgnevpm.o | . . . . . 6 β’ 1 = (1rβπ ) | |
34 | 17, 33 | zrh1 20936 | . . . . 5 β’ (π β Ring β (πβ1) = 1 ) |
35 | 34 | fveq2d 6850 | . . . 4 β’ (π β Ring β (πΌβ(πβ1)) = (πΌβ 1 )) |
36 | 27, 32, 35 | 3eqtr3d 2781 | . . 3 β’ (π β Ring β (πβ-1) = (πΌβ 1 )) |
37 | 36 | 3ad2ant1 1134 | . 2 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β (πβ-1) = (πΌβ 1 )) |
38 | 13, 16, 37 | 3eqtrd 2777 | 1 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β ((π β π)βπΉ) = (πΌβ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β cdif 3911 {cpr 4592 β ccom 5641 βΆwf 6496 βcfv 6500 (class class class)co 7361 Fincfn 8889 1c1 11060 -cneg 11394 β€cz 12507 Basecbs 17091 βΎs cress 17120 invgcminusg 18757 GrpHom cghm 19013 SymGrpcsymg 19156 pmSgncpsgn 19279 pmEvencevpm 19280 mulGrpcmgp 19904 1rcur 19921 Ringcrg 19972 RingHom crh 20153 βfldccnfld 20819 β€ringczring 20892 β€RHomczrh 20923 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-word 14412 df-lsw 14460 df-concat 14468 df-s1 14493 df-substr 14538 df-pfx 14568 df-splice 14647 df-reverse 14656 df-s2 14746 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-gsum 17332 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-efmnd 18687 df-grp 18759 df-minusg 18760 df-mulg 18881 df-subg 18933 df-ghm 19014 df-gim 19057 df-oppg 19132 df-symg 19157 df-pmtr 19232 df-psgn 19281 df-evpm 19282 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-rnghom 20156 df-drng 20221 df-subrg 20262 df-cnfld 20820 df-zring 20893 df-zrh 20927 |
This theorem is referenced by: mdetralt 21980 mdetunilem7 21990 |
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