MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gxmodid Structured version   Unicode version

Theorem gxmodid 23771
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 10655 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
2 nnrp 11005 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR+ )
3 modval 11715 . . . . . 6  |-  ( ( K  e.  RR  /\  M  e.  RR+ )  -> 
( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
41, 2, 3syl2an 477 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
543ad2ant2 1010 . . . 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 6112 . . 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 457 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  ZZ )
87zcnd 10753 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  CC )
9 nnz 10673 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  ZZ )
109adantl 466 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
11 nnre 10334 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  e.  RR )
12 nnne0 10359 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  =/=  0 )
13 redivcl 10055 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( K  /  M )  e.  RR )
141, 11, 12, 13syl3an 1260 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  NN  /\  M  e.  NN )  ->  ( K  /  M )  e.  RR )
15143anidm23 1277 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  /  M
)  e.  RR )
1615flcld 11653 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  ZZ )
1710, 16zmulcld 10758 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
1817zcnd 10753 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  CC )
198, 18negsubd 9730 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
20193ad2ant2 1010 . . . 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 6112 . . 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 988 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  G  e.  GrpOp )
23 simp3l 1016 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  A  e.  X
)
2417znegcld 10754 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
257, 24jca 532 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  e.  ZZ  /\  -u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ ) )
26253ad2ant2 1010 . . . 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 23767 . . . 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 1218 . . 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 2481 . 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 10753 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  CC )
3316zcnd 10753 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  CC )
3432, 33mulneg2d 9803 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  -u ( |_ `  ( K  /  M ) ) )  =  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )
35343ad2ant2 1010 . . . . 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 6112 . . . 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 1010 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  M  e.  ZZ )
3816znegcld 10754 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( |_ `  ( K  /  M
) )  e.  ZZ )
39383ad2ant2 1010 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  -u ( |_ `  ( K  /  M
) )  e.  ZZ )
4027, 28gxmul 23770 . . . . . 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 1222 . . . . 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 6103 . . . . . . 7  |-  ( ( A P M )  =  U  ->  (
( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P
-u ( |_ `  ( K  /  M
) ) ) )
4342adantl 466 . . . . . 6  |-  ( ( A  e.  X  /\  ( A P M )  =  U )  -> 
( ( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P -u ( |_
`  ( K  /  M ) ) ) )
44433ad2ant3 1011 . . . . 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 23765 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  -u ( |_ `  ( K  /  M ) )  e.  ZZ )  ->  ( U P -u ( |_
`  ( K  /  M ) ) )  =  U )
4722, 39, 46syl2anc 661 . . . . 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 2479 . . . 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 2477 . . 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 6112 . 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 1014 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  K  e.  ZZ )
5227, 28gxcl 23757 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
5322, 23, 51, 52syl3anc 1218 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P K )  e.  X
)
5427, 45grporid 23712 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A P K )  e.  X )  ->  (
( A P K ) G U )  =  ( A P K ) )
5522, 53, 54syl2anc 661 . 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 2479 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   ran crn 4846   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287    + caddc 9290    x. cmul 9292    - cmin 9600   -ucneg 9601    / cdiv 9998   NNcn 10327   ZZcz 10651   RR+crp 10996   |_cfl 11645    mod cmo 11713   GrpOpcgr 23678  GIdcgi 23679   ^gcgx 23682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fl 11647  df-mod 11714  df-seq 11812  df-grpo 23683  df-gid 23684  df-ginv 23685  df-gx 23687
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator