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Theorem gxmodid 21872
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 10291 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
2 nnrp 10626 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR+ )
3 modval 11257 . . . . . 6  |-  ( ( K  e.  RR  /\  M  e.  RR+ )  -> 
( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
41, 2, 3syl2an 465 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
543ad2ant2 980 . . . 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 6100 . . 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 445 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  ZZ )
87zcnd 10381 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  CC )
9 nnz 10308 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  ZZ )
109adantl 454 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
11 nnre 10012 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  e.  RR )
12 nnne0 10037 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  =/=  0 )
13 redivcl 9738 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( K  /  M )  e.  RR )
141, 11, 12, 13syl3an 1227 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  NN  /\  M  e.  NN )  ->  ( K  /  M )  e.  RR )
15143anidm23 1244 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  /  M
)  e.  RR )
1615flcld 11212 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  ZZ )
1710, 16zmulcld 10386 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
1817zcnd 10381 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  CC )
198, 18negsubd 9422 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
20193ad2ant2 980 . . . 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 6100 . . 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 958 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  G  e.  GrpOp )
23 simp3l 986 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  A  e.  X
)
2417znegcld 10382 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
257, 24jca 520 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  e.  ZZ  /\  -u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ ) )
26253ad2ant2 980 . . . 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 21868 . . . 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 1185 . . 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 2476 . 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 10381 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  CC )
3316zcnd 10381 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  CC )
3432, 33mulneg2d 9492 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  -u ( |_ `  ( K  /  M ) ) )  =  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )
35343ad2ant2 980 . . . . 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 6100 . . . 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 980 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  M  e.  ZZ )
3816znegcld 10382 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( |_ `  ( K  /  M
) )  e.  ZZ )
39383ad2ant2 980 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  -u ( |_ `  ( K  /  M
) )  e.  ZZ )
4027, 28gxmul 21871 . . . . . 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 1189 . . . . 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 6091 . . . . . . 7  |-  ( ( A P M )  =  U  ->  (
( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P
-u ( |_ `  ( K  /  M
) ) ) )
4342adantl 454 . . . . . 6  |-  ( ( A  e.  X  /\  ( A P M )  =  U )  -> 
( ( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P -u ( |_
`  ( K  /  M ) ) ) )
44433ad2ant3 981 . . . . 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 21866 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  -u ( |_ `  ( K  /  M ) )  e.  ZZ )  ->  ( U P -u ( |_
`  ( K  /  M ) ) )  =  U )
4722, 39, 46syl2anc 644 . . . . 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 2474 . . . 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 2472 . . 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 6100 . 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 984 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  K  e.  ZZ )
5227, 28gxcl 21858 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
5322, 23, 51, 52syl3anc 1185 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P K )  e.  X
)
5427, 45grporid 21813 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A P K )  e.  X )  ->  (
( A P K ) G U )  =  ( A P K ) )
5522, 53, 54syl2anc 644 . 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 2474 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   ran crn 4882   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297    / cdiv 9682   NNcn 10005   ZZcz 10287   RR+crp 10617   |_cfl 11206    mod cmo 11255   GrpOpcgr 21779  GIdcgi 21780   ^gcgx 21783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  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-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fl 11207  df-mod 11256  df-seq 11329  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gx 21788
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