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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 2728 | . . . . . 6 β’ (SymGrpβπ) = (SymGrpβπ) | |
2 | zrhpsgnevpm.s | . . . . . 6 β’ π = (pmSgnβπ) | |
3 | eqid 2728 | . . . . . 6 β’ ((mulGrpββfld) βΎs {1, -1}) = ((mulGrpββfld) βΎs {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 21520 | . . . . 5 β’ (π β Fin β π β ((SymGrpβπ) GrpHom ((mulGrpββfld) βΎs {1, -1}))) |
5 | zrhpsgnodpm.p | . . . . . 6 β’ π = (Baseβ(SymGrpβπ)) | |
6 | eqid 2728 | . . . . . 6 β’ (Baseβ((mulGrpββfld) βΎs {1, -1})) = (Baseβ((mulGrpββfld) βΎs {1, -1})) | |
7 | 5, 6 | ghmf 19181 | . . . . 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 1131 | . . 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 4127 | . . . 4 β’ (πΉ β (π β (pmEvenβπ)) β πΉ β π) | |
11 | 10 | 3ad2ant3 1132 | . . 3 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β πΉ β π) |
12 | fvco3 7002 | . . 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 582 | . 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 21527 | . . . 4 β’ ((π β Fin β§ πΉ β (π β (pmEvenβπ))) β (πβπΉ) = -1) |
15 | 14 | 3adant1 1127 | . . 3 β’ ((π β Ring β§ π β Fin β§ πΉ β (π β (pmEvenβπ))) β (πβπΉ) = -1) |
16 | 15 | fveq2d 6906 | . 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 21444 | . . . . . 6 β’ (π β Ring β π β (β€ring RingHom π )) |
19 | rhmghm 20430 | . . . . . 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 12630 | . . . . . 6 β’ 1 β β€ | |
22 | 21 | a1i 11 | . . . . 5 β’ (π β Ring β 1 β β€) |
23 | zringbas 21386 | . . . . . 6 β’ β€ = (Baseββ€ring) | |
24 | eqid 2728 | . . . . . 6 β’ (invgββ€ring) = (invgββ€ring) | |
25 | zrhpsgnodpm.i | . . . . . 6 β’ πΌ = (invgβπ ) | |
26 | 23, 24, 25 | ghminv 19184 | . . . . 5 β’ ((π β (β€ring GrpHom π ) β§ 1 β β€) β (πβ((invgββ€ring)β1)) = (πΌβ(πβ1))) |
27 | 20, 22, 26 | syl2anc 582 | . . . 4 β’ (π β Ring β (πβ((invgββ€ring)β1)) = (πΌβ(πβ1))) |
28 | zringinvg 21398 | . . . . . . . 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 2737 | . . . . . 6 β’ ((invgββ€ring)β1) = -1 |
31 | 30 | fveq2i 6905 | . . . . 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 21445 | . . . . 5 β’ (π β Ring β (πβ1) = 1 ) |
35 | 34 | fveq2d 6906 | . . . 4 β’ (π β Ring β (πΌβ(πβ1)) = (πΌβ 1 )) |
36 | 27, 32, 35 | 3eqtr3d 2776 | . . 3 β’ (π β Ring β (πβ-1) = (πΌβ 1 )) |
37 | 36 | 3ad2ant1 1130 | . 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 2772 | 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 1084 = wceq 1533 β wcel 2098 β cdif 3946 {cpr 4634 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 Fincfn 8970 1c1 11147 -cneg 11483 β€cz 12596 Basecbs 17187 βΎs cress 17216 invgcminusg 18898 GrpHom cghm 19174 SymGrpcsymg 19328 pmSgncpsgn 19451 pmEvencevpm 19452 mulGrpcmgp 20081 1rcur 20128 Ringcrg 20180 RingHom crh 20415 βfldccnfld 21286 β€ringczring 21379 β€RHomczrh 21432 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-word 14505 df-lsw 14553 df-concat 14561 df-s1 14586 df-substr 14631 df-pfx 14661 df-splice 14740 df-reverse 14749 df-s2 14839 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-gsum 17431 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-efmnd 18828 df-grp 18900 df-minusg 18901 df-mulg 19031 df-subg 19085 df-ghm 19175 df-gim 19220 df-oppg 19304 df-symg 19329 df-pmtr 19404 df-psgn 19453 df-evpm 19454 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-drng 20633 df-cnfld 21287 df-zring 21380 df-zrh 21436 |
This theorem is referenced by: mdetralt 22530 mdetunilem7 22540 |
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