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Theorem gxmodid 21058
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 10120 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
2 nnrp 10455 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR+ )
3 modval 11067 . . . . . 6  |-  ( ( K  e.  RR  /\  M  e.  RR+ )  -> 
( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
41, 2, 3syl2an 463 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  mod  M
)  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
543ad2ant2 977 . . . 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 5961 . . 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 443 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  ZZ )
87zcnd 10210 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  K  e.  CC )
9 nnz 10137 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  ZZ )
109adantl 452 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
11 nnre 9843 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  e.  RR )
12 nnne0 9868 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  =/=  0 )
13 redivcl 9569 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( K  /  M )  e.  RR )
141, 11, 12, 13syl3an 1224 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  NN  /\  M  e.  NN )  ->  ( K  /  M )  e.  RR )
15143anidm23 1241 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  /  M
)  e.  RR )
1615flcld 11022 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  ZZ )
1710, 16zmulcld 10215 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
1817zcnd 10210 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  CC )
198, 18negsubd 9253 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  +  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )  =  ( K  -  ( M  x.  ( |_ `  ( K  /  M ) ) ) ) )
20193ad2ant2 977 . . . 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 5961 . . 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 955 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  G  e.  GrpOp )
23 simp3l 983 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  A  e.  X
)
2417znegcld 10211 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ )
257, 24jca 518 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( K  e.  ZZ  /\  -u ( M  x.  ( |_ `  ( K  /  M ) ) )  e.  ZZ ) )
26253ad2ant2 977 . . . 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 21054 . . . 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 1182 . . 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 2396 . 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 10210 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  M  e.  CC )
3316zcnd 10210 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( |_ `  ( K  /  M ) )  e.  CC )
3432, 33mulneg2d 9323 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  ->  ( M  x.  -u ( |_ `  ( K  /  M ) ) )  =  -u ( M  x.  ( |_ `  ( K  /  M ) ) ) )
35343ad2ant2 977 . . . . 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 5961 . . . 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 977 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  M  e.  ZZ )
3816znegcld 10211 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  NN )  -> 
-u ( |_ `  ( K  /  M
) )  e.  ZZ )
39383ad2ant2 977 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  -u ( |_ `  ( K  /  M
) )  e.  ZZ )
4027, 28gxmul 21057 . . . . . 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 1186 . . . . 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 5952 . . . . . . 7  |-  ( ( A P M )  =  U  ->  (
( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P
-u ( |_ `  ( K  /  M
) ) ) )
4342adantl 452 . . . . . 6  |-  ( ( A  e.  X  /\  ( A P M )  =  U )  -> 
( ( A P M ) P -u ( |_ `  ( K  /  M ) ) )  =  ( U P -u ( |_
`  ( K  /  M ) ) ) )
44433ad2ant3 978 . . . . 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 21052 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  -u ( |_ `  ( K  /  M ) )  e.  ZZ )  ->  ( U P -u ( |_
`  ( K  /  M ) ) )  =  U )
4722, 39, 46syl2anc 642 . . . . 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 2394 . . . 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 2392 . . 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 5961 . 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 981 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  K  e.  ZZ )
5227, 28gxcl 21044 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
5322, 23, 51, 52syl3anc 1182 . . 3  |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U ) )  ->  ( A P K )  e.  X
)
5427, 45grporid 20999 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A P K )  e.  X )  ->  (
( A P K ) G U )  =  ( A P K ) )
5522, 53, 54syl2anc 642 . 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 2394 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   ran crn 4772   ` cfv 5337  (class class class)co 5945   RRcr 8826   0cc0 8827    + caddc 8830    x. cmul 8832    - cmin 9127   -ucneg 9128    / cdiv 9513   NNcn 9836   ZZcz 10116   RR+crp 10446   |_cfl 11016    mod cmo 11065   GrpOpcgr 20965  GIdcgi 20966   ^gcgx 20969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fl 11017  df-mod 11066  df-seq 11139  df-grpo 20970  df-gid 20971  df-ginv 20972  df-gx 20974
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