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Theorem gxmodid 21820
Description: Casting out powers of the identity element leaves the group power unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxmodid.1  |-  X  =  ran  G
gxmodid.2  |-  U  =  (GId `  G )
gxmodid.3  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxmodid  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( K  mod  M
) )  =  ( A P K ) )

Proof of Theorem gxmodid
StepHypRef Expression
1 zre 10242 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
2 nnrp 10577 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR+ )
3 modval 11207 . . . . . 6  |-  ( ( K  e.  RR  /\  M  e.  RR+ )  -> 
( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
41, 2, 3syl2an 464 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
543ad2ant2 979 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( K  mod  M )  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
65oveq2d 6056 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( K  mod  M
) )  =  ( A P ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) ) )
7 simpl 444 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  ZZ )
87zcnd 10332 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  CC )
9 nnz 10259 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  ZZ )
109adantl 453 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
11 nnre 9963 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  e.  RR )
12 nnne0 9988 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  =/=  0 )
13 redivcl 9689 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( K  /  M )  e.  RR )
141, 11, 12, 13syl3an 1226 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  NN  /\  M  e.  NN )  ->  ( K  /  M )  e.  RR )
15143anidm23 1243 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  /  M
)  e.  RR )
1615flcld 11162 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  ZZ )
1710, 16zmulcld 10337 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
1817zcnd 10332 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  CC )
198, 18negsubd 9373 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
20193ad2ant2 979 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
2120oveq2d 6056 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )  =  ( A P ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) ) )
22 simp1 957 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  G  e.  GrpOp )
23 simp3l 985 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  A  e.  X
)
2417znegcld 10333 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
257, 24jca 519 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  e.  ZZ  /\  -u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ ) )
26253ad2ant2 979 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( K  e.  ZZ  /\  -u ( M  x.  ( |_ `  ( K  /  M
) ) )  e.  ZZ ) )
27 gxmodid.1 . . . . 5  |-  X  =  ran  G
28 gxmodid.3 . . . . 5  |-  P  =  ( ^g `  G
)
2927, 28gxadd 21816 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u ( M  x.  ( |_ `  ( K  /  M
) ) )  e.  ZZ ) )  -> 
( A P ( K  +  -u ( M  x.  ( |_ `  ( K  /  M
) ) ) ) )  =  ( ( A P K ) G ( A P
-u ( M  x.  ( |_ `  ( K  /  M ) ) ) ) ) )
3022, 23, 26, 29syl3anc 1184 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )  =  ( ( A P K ) G ( A P -u ( M  x.  ( |_ `  ( K  /  M
) ) ) ) ) )
316, 21, 303eqtr2d 2442 . 2  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( K  mod  M
) )  =  ( ( A P K ) G ( A P -u ( M  x.  ( |_ `  ( K  /  M
) ) ) ) ) )
3210zcnd 10332 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  CC )
3316zcnd 10332 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  CC )
3432, 33mulneg2d 9443 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  -u ( |_ `  ( K  /  M ) ) )  =  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )
35343ad2ant2 979 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( M  x.  -u ( |_ `  ( K  /  M ) ) )  =  -u ( M  x.  ( |_ `  ( K  /  M
) ) ) )
3635oveq2d 6056 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( M  x.  -u ( |_ `  ( K  /  M ) ) ) )  =  ( A P -u ( M  x.  ( |_ `  ( K  /  M
) ) ) ) )
37103ad2ant2 979 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  M  e.  ZZ )
3816znegcld 10333 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( |_ `  ( K  /  M
) )  e.  ZZ )
39383ad2ant2 979 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  -u ( |_ `  ( K  /  M
) )  e.  ZZ )
4027, 28gxmul 21819 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( M  e.  ZZ  /\  -u ( |_ `  ( K  /  M ) )  e.  ZZ ) )  -> 
( A P ( M  x.  -u ( |_ `  ( K  /  M ) ) ) )  =  ( ( A P M ) P -u ( |_
`  ( K  /  M ) ) ) )
4122, 23, 37, 39, 40syl112anc 1188 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( M  x.  -u ( |_ `  ( K  /  M ) ) ) )  =  ( ( A P M ) P -u ( |_
`  ( K  /  M ) ) ) )
42 oveq1 6047 . . . . . . 7  |-  ( ( A P M )  =  U  ->  (
( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P
-u ( |_ `  ( K  /  M
) ) ) )
4342adantl 453 . . . . . 6  |-  ( ( A  e.  X  /\  ( A P M )  =  U )  -> 
( ( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P -u ( |_
`  ( K  /  M ) ) ) )
44433ad2ant3 980 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( ( A P M ) P
-u ( |_ `  ( K  /  M
) ) )  =  ( U P -u ( |_ `  ( K  /  M ) ) ) )
45 gxmodid.2 . . . . . . 7  |-  U  =  (GId `  G )
4645, 28gxid 21814 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  -u ( |_ `  ( K  /  M ) )  e.  ZZ )  ->  ( U P -u ( |_
`  ( K  /  M ) ) )  =  U )
4722, 39, 46syl2anc 643 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( U P
-u ( |_ `  ( K  /  M
) ) )  =  U )
4841, 44, 473eqtrd 2440 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( M  x.  -u ( |_ `  ( K  /  M ) ) ) )  =  U )
4936, 48eqtr3d 2438 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P
-u ( M  x.  ( |_ `  ( K  /  M ) ) ) )  =  U )
5049oveq2d 6056 . 2  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( ( A P K ) G ( A P -u ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )  =  ( ( A P K ) G U ) )
51 simp2l 983 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  K  e.  ZZ )
5227, 28gxcl 21806 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
5322, 23, 51, 52syl3anc 1184 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P K )  e.  X
)
5427, 45grporid 21761 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A P K )  e.  X )  ->  (
( A P K ) G U )  =  ( A P K ) )
5522, 53, 54syl2anc 643 . 2  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( ( A P K ) G U )  =  ( A P K ) )
5631, 50, 553eqtrd 2440 1  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P ( K  mod  M
) )  =  ( A P K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   ran crn 4838   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   ZZcz 10238   RR+crp 10568   |_cfl 11156    mod cmo 11205   GrpOpcgr 21727  GIdcgi 21728   ^gcgx 21731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gx 21736
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