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Mirrors > Home > MPE Home > Th. List > psgninv | Structured version Visualization version Unicode version |
Description: The sign of a permutation equals the sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
psgninv.s |
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psgninv.n |
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psgninv.p |
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Ref | Expression |
---|---|
psgninv |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgninv.s |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | psgninv.n |
. . . . 5
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3 | eqid 2451 |
. . . . 5
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4 | 1, 2, 3 | psgnghm2 19149 |
. . . 4
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5 | psgninv.p |
. . . . 5
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6 | eqid 2451 |
. . . . 5
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7 | eqid 2451 |
. . . . 5
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8 | 5, 6, 7 | ghminv 16890 |
. . . 4
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9 | 4, 8 | sylan 474 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 1, 5, 6 | symginv 17043 |
. . . . 5
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11 | 10 | adantl 468 |
. . . 4
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12 | 11 | fveq2d 5869 |
. . 3
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13 | eqid 2451 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | cnmsgnsubg 19145 |
. . . . 5
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15 | 3 | cnmsgnbas 19146 |
. . . . . . . 8
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16 | 5, 15 | ghmf 16887 |
. . . . . . 7
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17 | 4, 16 | syl 17 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | ffvelrnda 6022 |
. . . . 5
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19 | cnex 9620 |
. . . . . . . . 9
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20 | difss 3560 |
. . . . . . . . 9
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21 | 19, 20 | ssexi 4548 |
. . . . . . . 8
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22 | ax-1cn 9597 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() | |
23 | ax-1ne0 9608 |
. . . . . . . . . 10
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24 | eldifsn 4097 |
. . . . . . . . . 10
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25 | 22, 23, 24 | mpbir2an 931 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | neg1cn 10713 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() | |
27 | neg1ne0 10715 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() | |
28 | eldifsn 4097 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 26, 27, 28 | mpbir2an 931 |
. . . . . . . . 9
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30 | prssi 4128 |
. . . . . . . . 9
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31 | 25, 29, 30 | mp2an 678 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | ressabs 15188 |
. . . . . . . 8
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33 | 21, 31, 32 | mp2an 678 |
. . . . . . 7
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34 | 33 | eqcomi 2460 |
. . . . . 6
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35 | cnfldbas 18974 |
. . . . . . . 8
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36 | cnfld0 18992 |
. . . . . . . 8
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37 | cndrng 18997 |
. . . . . . . 8
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38 | 35, 36, 37 | drngui 17981 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | eqid 2451 |
. . . . . . 7
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40 | 38, 13, 39 | invrfval 17901 |
. . . . . 6
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41 | 34, 40, 7 | subginv 16824 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 14, 18, 41 | sylancr 669 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 31, 18 | sseldi 3430 |
. . . . . 6
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44 | eldifsn 4097 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
45 | 43, 44 | sylib 200 |
. . . . 5
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46 | cnfldinv 18999 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 45, 46 | syl 17 |
. . . 4
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48 | 42, 47 | eqtr3d 2487 |
. . 3
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49 | 9, 12, 48 | 3eqtr3d 2493 |
. 2
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50 | fvex 5875 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
51 | 50 | elpr 3986 |
. . . 4
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52 | 1div1e1 10300 |
. . . . . 6
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53 | oveq2 6298 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
54 | id 22 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
55 | 52, 53, 54 | 3eqtr4a 2511 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | divneg2 10331 |
. . . . . . . 8
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57 | 22, 22, 23, 56 | mp3an 1364 |
. . . . . . 7
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58 | 52 | negeqi 9868 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 57, 58 | eqtr3i 2475 |
. . . . . 6
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60 | oveq2 6298 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
61 | id 22 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
62 | 59, 60, 61 | 3eqtr4a 2511 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | 55, 62 | jaoi 381 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | 51, 63 | sylbi 199 |
. . 3
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65 | 18, 64 | syl 17 |
. 2
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66 | 49, 65 | eqtrd 2485 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-addf 9618 ax-mulf 9619 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-xor 1406 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-ot 3977 df-uni 4199 df-int 4235 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-se 4794 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-isom 5591 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-tpos 6973 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-2o 7183 df-oadd 7186 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-card 8373 df-cda 8598 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-3 10669 df-4 10670 df-5 10671 df-6 10672 df-7 10673 df-8 10674 df-9 10675 df-10 10676 df-n0 10870 df-z 10938 df-dec 11052 df-uz 11160 df-rp 11303 df-fz 11785 df-fzo 11916 df-seq 12214 df-exp 12273 df-hash 12516 df-word 12664 df-lsw 12665 df-concat 12666 df-s1 12667 df-substr 12668 df-splice 12669 df-reverse 12670 df-s2 12944 df-struct 15123 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-ress 15128 df-plusg 15203 df-mulr 15204 df-starv 15205 df-tset 15209 df-ple 15210 df-ds 15212 df-unif 15213 df-0g 15340 df-gsum 15341 df-mre 15492 df-mrc 15493 df-acs 15495 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-mhm 16582 df-submnd 16583 df-grp 16673 df-minusg 16674 df-subg 16814 df-ghm 16881 df-gim 16923 df-oppg 16997 df-symg 17019 df-pmtr 17083 df-psgn 17132 df-cmn 17432 df-abl 17433 df-mgp 17724 df-ur 17736 df-ring 17782 df-cring 17783 df-oppr 17851 df-dvdsr 17869 df-unit 17870 df-invr 17900 df-dvr 17911 df-drng 17977 df-cnfld 18971 |
This theorem is referenced by: zrhpsgninv 19153 evpmodpmf1o 19164 madjusmdetlem4 28656 |
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