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Theorem psgnuni 27401
Description: If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Hypotheses
Ref Expression
psgnuni.g  |-  G  =  ( SymGrp `  D )
psgnuni.t  |-  T  =  ran  (pmTrsp `  D
)
psgnuni.d  |-  ( ph  ->  D  e.  V )
psgnuni.w  |-  ( ph  ->  W  e. Word  T )
psgnuni.x  |-  ( ph  ->  X  e. Word  T )
psgnuni.e  |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )
Assertion
Ref Expression
psgnuni  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )

Proof of Theorem psgnuni
StepHypRef Expression
1 psgnuni.w . . . . . 6  |-  ( ph  ->  W  e. Word  T )
2 lencl 11737 . . . . . 6  |-  ( W  e. Word  T  ->  ( # `
 W )  e. 
NN0 )
31, 2syl 16 . . . . 5  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
43nn0zd 10375 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  ZZ )
5 m1expcl 11406 . . . 4  |-  ( (
# `  W )  e.  ZZ  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
64, 5syl 16 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
76zcnd 10378 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  CC )
8 psgnuni.x . . . . . 6  |-  ( ph  ->  X  e. Word  T )
9 lencl 11737 . . . . . 6  |-  ( X  e. Word  T  ->  ( # `
 X )  e. 
NN0 )
108, 9syl 16 . . . . 5  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
1110nn0zd 10375 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
12 m1expcl 11406 . . . 4  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1413zcnd 10378 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  CC )
15 neg1cn 10069 . . . 4  |-  -u 1  e.  CC
16 ax-1cn 9050 . . . . 5  |-  1  e.  CC
17 ax-1ne0 9061 . . . . 5  |-  1  =/=  0
1816, 17negne0i 9377 . . . 4  |-  -u 1  =/=  0
19 expne0i 11414 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( # `  X
)  e.  ZZ )  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
2015, 18, 19mp3an12 1270 . . 3  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
2111, 20syl 16 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
22 m1expaddsub 27400 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
234, 11, 22syl2anc 644 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( -u 1 ^ ( ( # `  W )  +  (
# `  X )
) ) )
24 expsub 11429 . . . . . 6  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  (
( # `  W )  e.  ZZ  /\  ( # `
 X )  e.  ZZ ) )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
2515, 18, 24mpanl12 665 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
264, 11, 25syl2anc 644 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
2723, 26eqtr3d 2472 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
28 revcl 11795 . . . . . . . 8  |-  ( X  e. Word  T  ->  (reverse `  X )  e. Word  T
)
298, 28syl 16 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  T )
30 ccatlen 11746 . . . . . . 7  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( # `
 ( W concat  (reverse `  X ) ) )  =  ( ( # `  W )  +  (
# `  (reverse `  X
) ) ) )
311, 29, 30syl2anc 644 . . . . . 6  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  (reverse `  X )
) ) )
32 revlen 11796 . . . . . . . 8  |-  ( X  e. Word  T  ->  ( # `
 (reverse `  X
) )  =  (
# `  X )
)
338, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (reverse `  X ) )  =  ( # `  X
) )
3433oveq2d 6099 . . . . . 6  |-  ( ph  ->  ( ( # `  W
)  +  ( # `  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3531, 34eqtrd 2470 . . . . 5  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3635oveq2d 6099 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
37 psgnuni.g . . . . 5  |-  G  =  ( SymGrp `  D )
38 psgnuni.t . . . . 5  |-  T  =  ran  (pmTrsp `  D
)
39 psgnuni.d . . . . 5  |-  ( ph  ->  D  e.  V )
40 ccatcl 11745 . . . . . 6  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( W concat  (reverse `  X )
)  e. Word  T )
411, 29, 40syl2anc 644 . . . . 5  |-  ( ph  ->  ( W concat  (reverse `  X
) )  e. Word  T
)
42 psgnuni.e . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )
4342fveq2d 5734 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  ( G  gsumg  W ) )  =  ( ( inv g `  G ) `  ( G  gsumg  X ) ) )
44 eqid 2438 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
4538, 37, 44symgtrinv 27392 . . . . . . . . . 10  |-  ( ( D  e.  V  /\  X  e. Word  T )  ->  ( ( inv g `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4639, 8, 45syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4743, 46eqtr2d 2471 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  (reverse `  X )
)  =  ( ( inv g `  G
) `  ( G  gsumg  W ) ) )
4847oveq2d 6099 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  W ) ) ) )
4937symggrp 15105 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
5039, 49syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
51 grpmnd 14819 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5250, 51syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
53 eqid 2438 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
5438, 37, 53symgtrf 27389 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
55 sswrd 11739 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
5654, 55ax-mp 8 . . . . . . . . . 10  |- Word  T  C_ Word  (
Base `  G )
5756, 1sseldi 3348 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
5853gsumwcl 14788 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
5952, 57, 58syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
60 eqid 2438 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
61 eqid 2438 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6253, 60, 61, 44grprinv 14854 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( G  gsumg  W )  e.  (
Base `  G )
)  ->  ( ( G  gsumg  W ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  W ) ) )  =  ( 0g `  G
) )
6350, 59, 62syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  W ) ) )  =  ( 0g `  G ) )
6448, 63eqtrd 2470 . . . . . 6  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( 0g `  G
) )
6556, 29sseldi 3348 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  ( Base `  G
) )
6653, 60gsumccat 14789 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
)  /\  (reverse `  X
)  e. Word  ( Base `  G ) )  -> 
( G  gsumg  ( W concat  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6752, 57, 65, 66syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6837symgid 15106 . . . . . . 7  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
6939, 68syl 16 . . . . . 6  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
7064, 67, 693eqtr4d 2480 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  (  _I  |`  D ) )
7137, 38, 39, 41, 70psgnunilem4 27399 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  1 )
7236, 71eqtr3d 2472 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  1 )
7327, 72eqtr3d 2472 . 2  |-  ( ph  ->  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) )  =  1 )
747, 14, 21, 73diveq1d 9800 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322    _I cid 4495   ran crn 4881    |` cres 4882   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    - cmin 9293   -ucneg 9294    / cdiv 9679   NN0cn0 10223   ZZcz 10284   ^cexp 11384   #chash 11620  Word cword 11719   concat cconcat 11720  reversecreverse 11724   Basecbs 13471   +g cplusg 13531   0gc0g 13725    gsumg cgsu 13726   Mndcmnd 14686   Grpcgrp 14687   inv gcminusg 14688   SymGrpcsymg 15094  pmTrspcpmtr 27363
This theorem is referenced by:  psgneu  27408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-xor 1315  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-reverse 11730  df-s2 11814  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-tset 13550  df-0g 13729  df-gsum 13730  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-subg 14943  df-ghm 15006  df-gim 15048  df-symg 15095  df-oppg 15144  df-pmtr 27364
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