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Theorem pgpfac1lem3a 16580
Description: Lemma for pgpfac1 16584. (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 3344 . . 3  |-  ( ph  ->  -.  C  e.  ( S  .(+)  W )
)
3 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
4 pgpprm 16095 . . . . . . . 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 16285 . . . . . . . . 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 16091 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
138, 9, 12syl2anc 661 . . . . . . 7  |-  ( ph  ->  E  e.  NN )
14 pceq0 13940 . . . . . . 7  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  (
( P  pCnt  E
)  =  0  <->  -.  P  ||  E ) )
155, 13, 14syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  E )  =  0  <->  -.  P  ||  E ) )
16 oveq2 6102 . . . . . 6  |-  ( ( P  pCnt  E )  =  0  ->  ( P ^ ( P  pCnt  E ) )  =  ( P ^ 0 ) )
1715, 16syl6bir 229 . . . . 5  |-  ( ph  ->  ( -.  P  ||  E  ->  ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 ) ) )
1810grpbn0 15570 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  B  =/=  (/) )
198, 18syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  (/) )
20 hashnncl 12137 . . . . . . . . . . . . 13  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
219, 20syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
2219, 21mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  e.  NN )
235, 22pccld 13920 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 B ) )  e.  NN0 )
2410, 11gexdvds3 16092 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  ||  ( # `  B ) )
258, 9, 24syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  E  ||  ( # `  B ) )
2610pgphash 16109 . . . . . . . . . . . 12  |-  ( ( P pGrp  G  /\  B  e.  Fin )  ->  ( # `
 B )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
273, 9, 26syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
2825, 27breqtrd 4319 . . . . . . . . . 10  |-  ( ph  ->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) )
29 oveq2 6102 . . . . . . . . . . . 12  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( P ^ k )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
3029breq2d 4307 . . . . . . . . . . 11  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( E  ||  ( P ^ k
)  <->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
3130rspcev 3076 . . . . . . . . . 10  |-  ( ( ( P  pCnt  ( # `
 B ) )  e.  NN0  /\  E  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) )  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
3223, 28, 31syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
33 pcprmpw2 13951 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  ( E. k  e.  NN0  E 
||  ( P ^
k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
345, 13, 33syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( E. k  e. 
NN0  E  ||  ( P ^ k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
3532, 34mpbid 210 . . . . . . . 8  |-  ( ph  ->  E  =  ( P ^ ( P  pCnt  E ) ) )
3635eqcomd 2448 . . . . . . 7  |-  ( ph  ->  ( P ^ ( P  pCnt  E ) )  =  E )
37 prmnn 13769 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
385, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
3938nncnd 10341 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
4039exp0d 12005 . . . . . . 7  |-  ( ph  ->  ( P ^ 0 )  =  1 )
4136, 40eqeq12d 2457 . . . . . 6  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
E  =  1 ) )
42 grpmnd 15553 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
438, 42syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
4410, 11gex1 16093 . . . . . . 7  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  B  ~~  1o ) )
4543, 44syl 16 . . . . . 6  |-  ( ph  ->  ( E  =  1  <-> 
B  ~~  1o )
)
4641, 45bitrd 253 . . . . 5  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
B  ~~  1o )
)
4717, 46sylibd 214 . . . 4  |-  ( ph  ->  ( -.  P  ||  E  ->  B  ~~  1o ) )
48 pgpfac1.s . . . . . . . . . . 11  |-  S  =  ( K `  { A } )
4910subgacs 15719 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
508, 49syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubGrp `  G )  e.  (ACS `  B )
)
5150acsmred 14597 . . . . . . . . . . . 12  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
52 pgpfac1.u . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5310subgss 15685 . . . . . . . . . . . . . 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 3360 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  B )
57 pgpfac1.k . . . . . . . . . . . . 13  |-  K  =  (mrCls `  (SubGrp `  G
) )
5857mrcsncl 14553 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
5951, 56, 58syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
6048, 59syl5eqel 2527 . . . . . . . . . 10  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
61 pgpfac1.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
62 pgpfac1.l . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  G )
6362lsmsubg2 16344 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
646, 60, 61, 63syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
65 pgpfac1.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
6665subg0cl 15692 . . . . . . . . 9  |-  ( ( S  .(+)  W )  e.  (SubGrp `  G )  ->  .0.  e.  ( S 
.(+)  W ) )
6764, 66syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  ( S 
.(+)  W ) )
6867snssd 4021 . . . . . . 7  |-  ( ph  ->  {  .0.  }  C_  ( S  .(+)  W ) )
6968adantr 465 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  {  .0.  } 
C_  ( S  .(+)  W ) )
701eldifad 3343 . . . . . . . . 9  |-  ( ph  ->  C  e.  U )
7154, 70sseldd 3360 . . . . . . . 8  |-  ( ph  ->  C  e.  B )
7271adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  B )
7310, 65grpidcl 15569 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .0.  e.  B )
748, 73syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
75 en1eqsn 7545 . . . . . . . 8  |-  ( (  .0.  e.  B  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7674, 75sylan 471 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7772, 76eleqtrd 2519 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e. 
{  .0.  } )
7869, 77sseldd 3360 . . . . 5  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  ( S  .(+)  W ) )
7978ex 434 . . . 4  |-  ( ph  ->  ( B  ~~  1o  ->  C  e.  ( S 
.(+)  W ) ) )
8047, 79syld 44 . . 3  |-  ( ph  ->  ( -.  P  ||  E  ->  C  e.  ( S  .(+)  W )
) )
812, 80mt3d 125 . 2  |-  ( ph  ->  P  ||  E )
82 pgpfac1.oe . . . . 5  |-  ( ph  ->  ( O `  A
)  =  E )
8313nncnd 10341 . . . . . 6  |-  ( ph  ->  E  e.  CC )
8438nnne0d 10369 . . . . . 6  |-  ( ph  ->  P  =/=  0 )
8583, 39, 84divcan1d 10111 . . . . 5  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  =  E )
8682, 85eqtr4d 2478 . . . 4  |-  ( ph  ->  ( O `  A
)  =  ( ( E  /  P )  x.  P ) )
87 nndivdvds 13544 . . . . . . . . . . . . 13  |-  ( ( E  e.  NN  /\  P  e.  NN )  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
8813, 38, 87syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
8981, 88mpbid 210 . . . . . . . . . . 11  |-  ( ph  ->  ( E  /  P
)  e.  NN )
9089nnzd 10749 . . . . . . . . . 10  |-  ( ph  ->  ( E  /  P
)  e.  ZZ )
91 pgpfac1.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
9290, 91zmulcld 10756 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  x.  M
)  e.  ZZ )
9356snssd 4021 . . . . . . . . . . . 12  |-  ( ph  ->  { A }  C_  B )
9451, 57, 93mrcssidd 14566 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
9594, 48syl6sseqr 3406 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  S )
96 snssg 4010 . . . . . . . . . . 11  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
9755, 96syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
9895, 97mpbird 232 . . . . . . . . 9  |-  ( ph  ->  A  e.  S )
99 pgpfac1.mg . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
10099subgmulgcl 15697 . . . . . . . . 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 1218 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  S )
102 prmz 13770 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1035, 102syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  ZZ )
10410, 99mulgcl 15647 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
1058, 103, 71, 104syl3anc 1218 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
10610, 99mulgcl 15647 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  A  e.  B )  ->  ( M  .x.  A )  e.  B )
1078, 91, 56, 106syl3anc 1218 . . . . . . . . . . 11  |-  ( ph  ->  ( M  .x.  A
)  e.  B )
108 eqid 2443 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
10910, 99, 108mulgdi 16317 . . . . . . . . . . 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 1220 . . . . . . . . . 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 6109 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( E 
.x.  C ) )
11210, 99mulgass 15660 . . . . . . . . . . . . 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 1220 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11410, 11, 99, 65gexid 16083 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( E  .x.  C )  =  .0.  )
11571, 114syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( E  .x.  C
)  =  .0.  )
116111, 113, 1153eqtr3rd 2484 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11710, 99mulgass 15660 . . . . . . . . . . . 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 1220 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  ( ( E  /  P ) 
.x.  ( M  .x.  A ) ) )
119116, 118oveq12d 6112 . . . . . . . . . 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 15685 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  B
)
12160, 120syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  B )
122121, 101sseldd 3360 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )
12310, 108, 65grplid 15571 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  (  .0.  ( +g  `  G ) ( ( ( E  /  P
)  x.  M ) 
.x.  A ) )  =  ( ( ( E  /  P )  x.  M )  .x.  A ) )
125110, 119, 1243eqtr2d 2481 . . . . . . . . 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 15697 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  e.  W )
129125, 128eqeltrrd 2518 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  W )
130101, 129elind 3543 . . . . . . 7  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  ( S  i^i  W ) )
131 pgpfac1.i . . . . . . 7  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
132130, 131eleqtrd 2519 . . . . . 6  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  {  .0.  } )
133 elsni 3905 . . . . . 6  |-  ( ( ( ( E  /  P )  x.  M
)  .x.  A )  e.  {  .0.  }  ->  ( ( ( E  /  P )  x.  M
)  .x.  A )  =  .0.  )
134132, 133syl 16 . . . . 5  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
135 pgpfac1.o . . . . . . 7  |-  O  =  ( od `  G
)
13610, 135, 99, 65oddvds 16053 . . . . . 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.  ) )
1378, 56, 92, 136syl3anc 1218 . . . . 5  |-  ( ph  ->  ( ( O `  A )  ||  (
( E  /  P
)  x.  M )  <-> 
( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
)
138134, 137mpbird 232 . . . 4  |-  ( ph  ->  ( O `  A
)  ||  ( ( E  /  P )  x.  M ) )
13986, 138eqbrtrrd 4317 . . 3  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  ||  ( ( E  /  P )  x.  M ) )
14089nnne0d 10369 . . . 4  |-  ( ph  ->  ( E  /  P
)  =/=  0 )
141 dvdscmulr 13564 . . . 4  |-  ( ( P  e.  ZZ  /\  M  e.  ZZ  /\  (
( E  /  P
)  e.  ZZ  /\  ( E  /  P
)  =/=  0 ) )  ->  ( (
( E  /  P
)  x.  P ) 
||  ( ( E  /  P )  x.  M )  <->  P  ||  M
) )
142103, 91, 90, 140, 141syl112anc 1222 . . 3  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  ||  (
( E  /  P
)  x.  M )  <-> 
P  ||  M )
)
143139, 142mpbid 210 . 2  |-  ( ph  ->  P  ||  M )
14481, 143jca 532 1  |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609   A.wral 2718   E.wrex 2719    \ cdif 3328    i^i cin 3330    C_ wss 3331    C. wpss 3332   (/)c0 3640   {csn 3880   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   1oc1o 6916    ~~ cen 7310   Fincfn 7313   0cc0 9285   1c1 9286    x. cmul 9290    / cdiv 9996   NNcn 10325   NN0cn0 10582   ZZcz 10649   ^cexp 11868   #chash 12106    || cdivides 13538   Primecprime 13766    pCnt cpc 13906   Basecbs 14177   +g cplusg 14241   0gc0g 14381  Moorecmre 14523  mrClscmrc 14524  ACScacs 14526   Mndcmnd 15412   Grpcgrp 15413  .gcmg 15417  SubGrpcsubg 15678   odcod 16031  gExcgex 16032   pGrp cpgp 16033   LSSumclsm 16136   Abelcabel 16281
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 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-disj 4266  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-omul 6928  df-er 7104  df-ec 7106  df-qs 7110  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-acn 8115  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-fac 12055  df-bc 12082  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-sum 13167  df-dvds 13539  df-gcd 13694  df-prm 13767  df-pc 13907  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-0g 14383  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-submnd 15468  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-eqg 15683  df-ga 15811  df-cntz 15838  df-od 16035  df-gex 16036  df-pgp 16037  df-lsm 16138  df-cmn 16282  df-abl 16283
This theorem is referenced by:  pgpfac1lem3  16581
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