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Theorem gxmodid 21828
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 10250 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
2 nnrp 10585 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR+ )
3 modval 11215 . . . . . 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 6064 . . 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 10340 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  CC )
9 nnz 10267 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  ZZ )
109adantl 453 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
11 nnre 9971 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  e.  RR )
12 nnne0 9996 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  =/=  0 )
13 redivcl 9697 . . . . . . . . . . 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 11170 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  ZZ )
1710, 16zmulcld 10345 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
1817zcnd 10340 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  CC )
198, 18negsubd 9381 . . . . 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 6064 . . 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 10341 . . . . . 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 21824 . . . 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 2450 . 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 10340 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  CC )
3316zcnd 10340 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  CC )
3432, 33mulneg2d 9451 . . . . . 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 6064 . . . 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 10341 . . . . . . 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 21827 . . . . . 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 6055 . . . . . . 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 21822 . . . . . 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 2448 . . . 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 2446 . . 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 6064 . 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 21814 . . . 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 21769 . . 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 2448 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 2575   ran crn 4846   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954    + caddc 8957    x. cmul 8959    - cmin 9255   -ucneg 9256    / cdiv 9641   NNcn 9964   ZZcz 10246   RR+crp 10576   |_cfl 11164    mod cmo 11213   GrpOpcgr 21735  GIdcgi 21736   ^gcgx 21739
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fl 11165  df-mod 11214  df-seq 11287  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gx 21744
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