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Theorem pgpfac1lem3a 15561
Description: Lemma for pgpfac1 15565. (Contributed by Mario Carneiro, 4-Jun-2016.)
Hypotheses
Ref Expression
pgpfac1.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
pgpfac1.s  |-  S  =  ( K `  { A } )
pgpfac1.b  |-  B  =  ( Base `  G
)
pgpfac1.o  |-  O  =  ( od `  G
)
pgpfac1.e  |-  E  =  (gEx `  G )
pgpfac1.z  |-  .0.  =  ( 0g `  G )
pgpfac1.l  |-  .(+)  =  (
LSSum `  G )
pgpfac1.p  |-  ( ph  ->  P pGrp  G )
pgpfac1.g  |-  ( ph  ->  G  e.  Abel )
pgpfac1.n  |-  ( ph  ->  B  e.  Fin )
pgpfac1.oe  |-  ( ph  ->  ( O `  A
)  =  E )
pgpfac1.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac1.au  |-  ( ph  ->  A  e.  U )
pgpfac1.w  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
pgpfac1.i  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
pgpfac1.ss  |-  ( ph  ->  ( S  .(+)  W ) 
C_  U )
pgpfac1.2  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )
pgpfac1.c  |-  ( ph  ->  C  e.  ( U 
\  ( S  .(+)  W ) ) )
pgpfac1.mg  |-  .x.  =  (.g
`  G )
pgpfac1.m  |-  ( ph  ->  M  e.  ZZ )
pgpfac1.mw  |-  ( ph  ->  ( ( P  .x.  C ) ( +g  `  G ) ( M 
.x.  A ) )  e.  W )
Assertion
Ref Expression
pgpfac1lem3a  |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
Distinct variable groups:    w, A    w, 
.(+)    w, P    w, G    w, U    w, C    w, S    w, W    ph, w    w,  .x.    w, K
Allowed substitution hints:    B( w)    E( w)    M( w)    O( w)    .0. (
w)

Proof of Theorem pgpfac1lem3a
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 pgpfac1.c . . . 4  |-  ( ph  ->  C  e.  ( U 
\  ( S  .(+)  W ) ) )
21eldifbd 3276 . . 3  |-  ( ph  ->  -.  C  e.  ( S  .(+)  W )
)
3 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
4 pgpprm 15154 . . . . . . . 8  |-  ( P pGrp 
G  ->  P  e.  Prime )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  Prime )
6 pgpfac1.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 15344 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
9 pgpfac1.n . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
10 pgpfac1.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
11 pgpfac1.e . . . . . . . . 9  |-  E  =  (gEx `  G )
1210, 11gexcl2 15150 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
138, 9, 12syl2anc 643 . . . . . . 7  |-  ( ph  ->  E  e.  NN )
14 pceq0 13171 . . . . . . 7  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  (
( P  pCnt  E
)  =  0  <->  -.  P  ||  E ) )
155, 13, 14syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  E )  =  0  <->  -.  P  ||  E ) )
16 oveq2 6028 . . . . . 6  |-  ( ( P  pCnt  E )  =  0  ->  ( P ^ ( P  pCnt  E ) )  =  ( P ^ 0 ) )
1715, 16syl6bir 221 . . . . 5  |-  ( ph  ->  ( -.  P  ||  E  ->  ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 ) ) )
1810grpbn0 14761 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  B  =/=  (/) )
198, 18syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  (/) )
20 hashnncl 11572 . . . . . . . . . . . . 13  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
219, 20syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
2219, 21mpbird 224 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  e.  NN )
235, 22pccld 13151 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 B ) )  e.  NN0 )
2410, 11gexdvds3 15151 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  ||  ( # `  B ) )
258, 9, 24syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  E  ||  ( # `  B ) )
2610pgphash 15168 . . . . . . . . . . . 12  |-  ( ( P pGrp  G  /\  B  e.  Fin )  ->  ( # `
 B )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
273, 9, 26syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
2825, 27breqtrd 4177 . . . . . . . . . 10  |-  ( ph  ->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) )
29 oveq2 6028 . . . . . . . . . . . 12  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( P ^ k )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
3029breq2d 4165 . . . . . . . . . . 11  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( E  ||  ( P ^ k
)  <->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
3130rspcev 2995 . . . . . . . . . 10  |-  ( ( ( P  pCnt  ( # `
 B ) )  e.  NN0  /\  E  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) )  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
3223, 28, 31syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
33 pcprmpw2 13182 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  ( E. k  e.  NN0  E 
||  ( P ^
k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
345, 13, 33syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( E. k  e. 
NN0  E  ||  ( P ^ k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
3532, 34mpbid 202 . . . . . . . 8  |-  ( ph  ->  E  =  ( P ^ ( P  pCnt  E ) ) )
3635eqcomd 2392 . . . . . . 7  |-  ( ph  ->  ( P ^ ( P  pCnt  E ) )  =  E )
37 prmnn 13009 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
385, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
3938nncnd 9948 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
4039exp0d 11444 . . . . . . 7  |-  ( ph  ->  ( P ^ 0 )  =  1 )
4136, 40eqeq12d 2401 . . . . . 6  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
E  =  1 ) )
42 grpmnd 14744 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
438, 42syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
4410, 11gex1 15152 . . . . . . 7  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  B  ~~  1o ) )
4543, 44syl 16 . . . . . 6  |-  ( ph  ->  ( E  =  1  <-> 
B  ~~  1o )
)
4641, 45bitrd 245 . . . . 5  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
B  ~~  1o )
)
4717, 46sylibd 206 . . . 4  |-  ( ph  ->  ( -.  P  ||  E  ->  B  ~~  1o ) )
48 pgpfac1.s . . . . . . . . . . 11  |-  S  =  ( K `  { A } )
4910subgacs 14902 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
508, 49syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubGrp `  G )  e.  (ACS `  B )
)
5150acsmred 13808 . . . . . . . . . . . 12  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
52 pgpfac1.u . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5310subgss 14872 . . . . . . . . . . . . . 14  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
5452, 53syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  U  C_  B )
55 pgpfac1.au . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  U )
5654, 55sseldd 3292 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  B )
57 pgpfac1.k . . . . . . . . . . . . 13  |-  K  =  (mrCls `  (SubGrp `  G
) )
5857mrcsncl 13764 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
5951, 56, 58syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
6048, 59syl5eqel 2471 . . . . . . . . . 10  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
61 pgpfac1.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
62 pgpfac1.l . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  G )
6362lsmsubg2 15401 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
646, 60, 61, 63syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
65 pgpfac1.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
6665subg0cl 14879 . . . . . . . . 9  |-  ( ( S  .(+)  W )  e.  (SubGrp `  G )  ->  .0.  e.  ( S 
.(+)  W ) )
6764, 66syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  ( S 
.(+)  W ) )
6867snssd 3886 . . . . . . 7  |-  ( ph  ->  {  .0.  }  C_  ( S  .(+)  W ) )
6968adantr 452 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  {  .0.  } 
C_  ( S  .(+)  W ) )
701eldifad 3275 . . . . . . . . 9  |-  ( ph  ->  C  e.  U )
7154, 70sseldd 3292 . . . . . . . 8  |-  ( ph  ->  C  e.  B )
7271adantr 452 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  B )
7310, 65grpidcl 14760 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .0.  e.  B )
748, 73syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
75 en1eqsn 7274 . . . . . . . 8  |-  ( (  .0.  e.  B  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7674, 75sylan 458 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7772, 76eleqtrd 2463 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e. 
{  .0.  } )
7869, 77sseldd 3292 . . . . 5  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  ( S  .(+)  W ) )
7978ex 424 . . . 4  |-  ( ph  ->  ( B  ~~  1o  ->  C  e.  ( S 
.(+)  W ) ) )
8047, 79syld 42 . . 3  |-  ( ph  ->  ( -.  P  ||  E  ->  C  e.  ( S  .(+)  W )
) )
812, 80mt3d 119 . 2  |-  ( ph  ->  P  ||  E )
82 pgpfac1.oe . . . . 5  |-  ( ph  ->  ( O `  A
)  =  E )
8313nncnd 9948 . . . . . 6  |-  ( ph  ->  E  e.  CC )
8438nnne0d 9976 . . . . . 6  |-  ( ph  ->  P  =/=  0 )
8583, 39, 84divcan1d 9723 . . . . 5  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  =  E )
8682, 85eqtr4d 2422 . . . 4  |-  ( ph  ->  ( O `  A
)  =  ( ( E  /  P )  x.  P ) )
87 nndivdvds 12785 . . . . . . . . . . . . 13  |-  ( ( E  e.  NN  /\  P  e.  NN )  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
8813, 38, 87syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
8981, 88mpbid 202 . . . . . . . . . . 11  |-  ( ph  ->  ( E  /  P
)  e.  NN )
9089nnzd 10306 . . . . . . . . . 10  |-  ( ph  ->  ( E  /  P
)  e.  ZZ )
91 pgpfac1.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
9290, 91zmulcld 10313 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  x.  M
)  e.  ZZ )
9356snssd 3886 . . . . . . . . . . . 12  |-  ( ph  ->  { A }  C_  B )
9451, 57, 93mrcssidd 13777 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
9594, 48syl6sseqr 3338 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  S )
96 snssg 3875 . . . . . . . . . . 11  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
9755, 96syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
9895, 97mpbird 224 . . . . . . . . 9  |-  ( ph  ->  A  e.  S )
99 pgpfac1.mg . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
10099subgmulgcl 14884 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  (
( E  /  P
)  x.  M )  e.  ZZ  /\  A  e.  S )  ->  (
( ( E  /  P )  x.  M
)  .x.  A )  e.  S )
10160, 92, 98, 100syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  S )
102 prmz 13010 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1035, 102syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  ZZ )
10410, 99mulgcl 14834 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
1058, 103, 71, 104syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
10610, 99mulgcl 14834 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  A  e.  B )  ->  ( M  .x.  A )  e.  B )
1078, 91, 56, 106syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( M  .x.  A
)  e.  B )
108 eqid 2387 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
10910, 99, 108mulgdi 15376 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  (
( E  /  P
)  e.  ZZ  /\  ( P  .x.  C )  e.  B  /\  ( M  .x.  A )  e.  B ) )  -> 
( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  =  ( ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) ( +g  `  G ) ( ( E  /  P )  .x.  ( M  .x.  A ) ) ) )
1106, 90, 105, 107, 109syl13anc 1186 . . . . . . . . . 10  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  =  ( ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) ( +g  `  G ) ( ( E  /  P )  .x.  ( M  .x.  A ) ) ) )
11185oveq1d 6035 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( E 
.x.  C ) )
11210, 99mulgass 14847 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  ( ( E  /  P )  e.  ZZ  /\  P  e.  ZZ  /\  C  e.  B )
)  ->  ( (
( E  /  P
)  x.  P ) 
.x.  C )  =  ( ( E  /  P )  .x.  ( P  .x.  C ) ) )
1138, 90, 103, 71, 112syl13anc 1186 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11410, 11, 99, 65gexid 15142 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( E  .x.  C )  =  .0.  )
11571, 114syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( E  .x.  C
)  =  .0.  )
116111, 113, 1153eqtr3rd 2428 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11710, 99mulgass 14847 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( ( E  /  P )  e.  ZZ  /\  M  e.  ZZ  /\  A  e.  B )
)  ->  ( (
( E  /  P
)  x.  M ) 
.x.  A )  =  ( ( E  /  P )  .x.  ( M  .x.  A ) ) )
1188, 90, 91, 56, 117syl13anc 1186 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  ( ( E  /  P ) 
.x.  ( M  .x.  A ) ) )
119116, 118oveq12d 6038 . . . . . . . . . 10  |-  ( ph  ->  (  .0.  ( +g  `  G ) ( ( ( E  /  P
)  x.  M ) 
.x.  A ) )  =  ( ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) ( +g  `  G ) ( ( E  /  P )  .x.  ( M  .x.  A ) ) ) )
12010subgss 14872 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  B
)
12160, 120syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  B )
122121, 101sseldd 3292 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )
12310, 108, 65grplid 14762 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )  ->  (  .0.  ( +g  `  G ) ( ( ( E  /  P )  x.  M
)  .x.  A )
)  =  ( ( ( E  /  P
)  x.  M ) 
.x.  A ) )
1248, 122, 123syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  (  .0.  ( +g  `  G ) ( ( ( E  /  P
)  x.  M ) 
.x.  A ) )  =  ( ( ( E  /  P )  x.  M )  .x.  A ) )
125110, 119, 1243eqtr2d 2425 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  =  ( ( ( E  /  P )  x.  M )  .x.  A ) )
126 pgpfac1.mw . . . . . . . . . 10  |-  ( ph  ->  ( ( P  .x.  C ) ( +g  `  G ) ( M 
.x.  A ) )  e.  W )
12799subgmulgcl 14884 . . . . . . . . . 10  |-  ( ( W  e.  (SubGrp `  G )  /\  ( E  /  P )  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( M  .x.  A ) )  e.  W )  ->  (
( E  /  P
)  .x.  ( ( P  .x.  C ) ( +g  `  G ) ( M  .x.  A
) ) )  e.  W )
12861, 90, 126, 127syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  e.  W )
129125, 128eqeltrrd 2462 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  W )
130 elin 3473 . . . . . . . 8  |-  ( ( ( ( E  /  P )  x.  M
)  .x.  A )  e.  ( S  i^i  W
)  <->  ( ( ( ( E  /  P
)  x.  M ) 
.x.  A )  e.  S  /\  ( ( ( E  /  P
)  x.  M ) 
.x.  A )  e.  W ) )
131101, 129, 130sylanbrc 646 . . . . . . 7  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  ( S  i^i  W ) )
132 pgpfac1.i . . . . . . 7  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
133131, 132eleqtrd 2463 . . . . . 6  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  {  .0.  } )
134 elsni 3781 . . . . . 6  |-  ( ( ( ( E  /  P )  x.  M
)  .x.  A )  e.  {  .0.  }  ->  ( ( ( E  /  P )  x.  M
)  .x.  A )  =  .0.  )
135133, 134syl 16 . . . . 5  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
136 pgpfac1.o . . . . . . 7  |-  O  =  ( od `  G
)
13710, 136, 99, 65oddvds 15112 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  B  /\  ( ( E  /  P )  x.  M
)  e.  ZZ )  ->  ( ( O `
 A )  ||  ( ( E  /  P )  x.  M
)  <->  ( ( ( E  /  P )  x.  M )  .x.  A )  =  .0.  ) )
1388, 56, 92, 137syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( O `  A )  ||  (
( E  /  P
)  x.  M )  <-> 
( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
)
139135, 138mpbird 224 . . . 4  |-  ( ph  ->  ( O `  A
)  ||  ( ( E  /  P )  x.  M ) )
14086, 139eqbrtrrd 4175 . . 3  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  ||  ( ( E  /  P )  x.  M ) )
14189nnne0d 9976 . . . 4  |-  ( ph  ->  ( E  /  P
)  =/=  0 )
142 dvdscmulr 12805 . . . 4  |-  ( ( P  e.  ZZ  /\  M  e.  ZZ  /\  (
( E  /  P
)  e.  ZZ  /\  ( E  /  P
)  =/=  0 ) )  ->  ( (
( E  /  P
)  x.  P ) 
||  ( ( E  /  P )  x.  M )  <->  P  ||  M
) )
143103, 91, 90, 141, 142syl112anc 1188 . . 3  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  ||  (
( E  /  P
)  x.  M )  <-> 
P  ||  M )
)
144140, 143mpbid 202 . 2  |-  ( ph  ->  P  ||  M )
14581, 144jca 519 1  |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650    \ cdif 3260    i^i cin 3262    C_ wss 3263    C. wpss 3264   (/)c0 3571   {csn 3757   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   1oc1o 6653    ~~ cen 7042   Fincfn 7045   0cc0 8923   1c1 8924    x. cmul 8928    / cdiv 9609   NNcn 9932   NN0cn0 10153   ZZcz 10214   ^cexp 11309   #chash 11545    || cdivides 12779   Primecprime 13006    pCnt cpc 13137   Basecbs 13396   +g cplusg 13456   0gc0g 13650  Moorecmre 13734  mrClscmrc 13735  ACScacs 13737   Mndcmnd 14611   Grpcgrp 14612  .gcmg 14616  SubGrpcsubg 14865   odcod 15090  gExcgex 15091   pGrp cpgp 15092   LSSumclsm 15195   Abelcabel 15340
This theorem is referenced by:  pgpfac1lem3  15562
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6841  df-ec 6843  df-qs 6847  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-acn 7762  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-eqg 14870  df-ga 14994  df-cntz 15043  df-od 15094  df-gex 15095  df-pgp 15096  df-lsm 15197  df-cmn 15341  df-abl 15342
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