MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpfac1lem3a Structured version   Unicode version

Theorem pgpfac1lem3a 17644
Description: Lemma for pgpfac1 17648. (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 3455 . . 3  |-  ( ph  ->  -.  C  e.  ( S  .(+)  W )
)
3 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
4 pgpprm 17180 . . . . . . . 8  |-  ( P pGrp 
G  ->  P  e.  Prime )
53, 4syl 17 . . . . . . 7  |-  ( ph  ->  P  e.  Prime )
6 pgpfac1.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 17370 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
86, 7syl 17 . . . . . . . 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 17176 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
138, 9, 12syl2anc 665 . . . . . . 7  |-  ( ph  ->  E  e.  NN )
14 pceq0 14783 . . . . . . 7  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  (
( P  pCnt  E
)  =  0  <->  -.  P  ||  E ) )
155, 13, 14syl2anc 665 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  E )  =  0  <->  -.  P  ||  E ) )
16 oveq2 6313 . . . . . 6  |-  ( ( P  pCnt  E )  =  0  ->  ( P ^ ( P  pCnt  E ) )  =  ( P ^ 0 ) )
1715, 16syl6bir 232 . . . . 5  |-  ( ph  ->  ( -.  P  ||  E  ->  ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 ) ) )
1810grpbn0 16646 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  B  =/=  (/) )
198, 18syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  (/) )
20 hashnncl 12544 . . . . . . . . . . . . 13  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
219, 20syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
2219, 21mpbird 235 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  e.  NN )
235, 22pccld 14763 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 B ) )  e.  NN0 )
2410, 11gexdvds3 17177 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  ||  ( # `  B ) )
258, 9, 24syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  E  ||  ( # `  B ) )
2610pgphash 17194 . . . . . . . . . . . 12  |-  ( ( P pGrp  G  /\  B  e.  Fin )  ->  ( # `
 B )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
273, 9, 26syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
2825, 27breqtrd 4450 . . . . . . . . . 10  |-  ( ph  ->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) )
29 oveq2 6313 . . . . . . . . . . . 12  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( P ^ k )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
3029breq2d 4438 . . . . . . . . . . 11  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( E  ||  ( P ^ k
)  <->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
3130rspcev 3188 . . . . . . . . . 10  |-  ( ( ( P  pCnt  ( # `
 B ) )  e.  NN0  /\  E  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) )  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
3223, 28, 31syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
33 pcprmpw2 14794 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  ( E. k  e.  NN0  E 
||  ( P ^
k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
345, 13, 33syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( E. k  e. 
NN0  E  ||  ( P ^ k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
3532, 34mpbid 213 . . . . . . . 8  |-  ( ph  ->  E  =  ( P ^ ( P  pCnt  E ) ) )
3635eqcomd 2437 . . . . . . 7  |-  ( ph  ->  ( P ^ ( P  pCnt  E ) )  =  E )
37 prmnn 14596 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
385, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
3938nncnd 10625 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
4039exp0d 12407 . . . . . . 7  |-  ( ph  ->  ( P ^ 0 )  =  1 )
4136, 40eqeq12d 2451 . . . . . 6  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
E  =  1 ) )
42 grpmnd 16629 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
438, 42syl 17 . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
4410, 11gex1 17178 . . . . . . 7  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  B  ~~  1o ) )
4543, 44syl 17 . . . . . 6  |-  ( ph  ->  ( E  =  1  <-> 
B  ~~  1o )
)
4641, 45bitrd 256 . . . . 5  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
B  ~~  1o )
)
4717, 46sylibd 217 . . . 4  |-  ( ph  ->  ( -.  P  ||  E  ->  B  ~~  1o ) )
48 pgpfac1.s . . . . . . . . . . 11  |-  S  =  ( K `  { A } )
4910subgacs 16803 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
508, 49syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubGrp `  G )  e.  (ACS `  B )
)
5150acsmred 15513 . . . . . . . . . . . 12  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
52 pgpfac1.u . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5310subgss 16769 . . . . . . . . . . . . . 14  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
5452, 53syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  U  C_  B )
55 pgpfac1.au . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  U )
5654, 55sseldd 3471 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  B )
57 pgpfac1.k . . . . . . . . . . . . 13  |-  K  =  (mrCls `  (SubGrp `  G
) )
5857mrcsncl 15469 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
5951, 56, 58syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
6048, 59syl5eqel 2521 . . . . . . . . . 10  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
61 pgpfac1.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
62 pgpfac1.l . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  G )
6362lsmsubg2 17432 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
646, 60, 61, 63syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
65 pgpfac1.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
6665subg0cl 16776 . . . . . . . . 9  |-  ( ( S  .(+)  W )  e.  (SubGrp `  G )  ->  .0.  e.  ( S 
.(+)  W ) )
6764, 66syl 17 . . . . . . . 8  |-  ( ph  ->  .0.  e.  ( S 
.(+)  W ) )
6867snssd 4148 . . . . . . 7  |-  ( ph  ->  {  .0.  }  C_  ( S  .(+)  W ) )
6968adantr 466 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  {  .0.  } 
C_  ( S  .(+)  W ) )
701eldifad 3454 . . . . . . . . 9  |-  ( ph  ->  C  e.  U )
7154, 70sseldd 3471 . . . . . . . 8  |-  ( ph  ->  C  e.  B )
7271adantr 466 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  B )
7310, 65grpidcl 16645 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .0.  e.  B )
748, 73syl 17 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
75 en1eqsn 7807 . . . . . . . 8  |-  ( (  .0.  e.  B  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7674, 75sylan 473 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7772, 76eleqtrd 2519 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e. 
{  .0.  } )
7869, 77sseldd 3471 . . . . 5  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  ( S  .(+)  W ) )
7978ex 435 . . . 4  |-  ( ph  ->  ( B  ~~  1o  ->  C  e.  ( S 
.(+)  W ) ) )
8047, 79syld 45 . . 3  |-  ( ph  ->  ( -.  P  ||  E  ->  C  e.  ( S  .(+)  W )
) )
812, 80mt3d 128 . 2  |-  ( ph  ->  P  ||  E )
82 pgpfac1.oe . . . . 5  |-  ( ph  ->  ( O `  A
)  =  E )
8313nncnd 10625 . . . . . 6  |-  ( ph  ->  E  e.  CC )
8438nnne0d 10654 . . . . . 6  |-  ( ph  ->  P  =/=  0 )
8583, 39, 84divcan1d 10383 . . . . 5  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  =  E )
8682, 85eqtr4d 2473 . . . 4  |-  ( ph  ->  ( O `  A
)  =  ( ( E  /  P )  x.  P ) )
87 nndivdvds 14289 . . . . . . . . . . . . 13  |-  ( ( E  e.  NN  /\  P  e.  NN )  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
8813, 38, 87syl2anc 665 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
8981, 88mpbid 213 . . . . . . . . . . 11  |-  ( ph  ->  ( E  /  P
)  e.  NN )
9089nnzd 11039 . . . . . . . . . 10  |-  ( ph  ->  ( E  /  P
)  e.  ZZ )
91 pgpfac1.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
9290, 91zmulcld 11046 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  x.  M
)  e.  ZZ )
9356snssd 4148 . . . . . . . . . . . 12  |-  ( ph  ->  { A }  C_  B )
9451, 57, 93mrcssidd 15482 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
9594, 48syl6sseqr 3517 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  S )
96 snssg 4136 . . . . . . . . . . 11  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
9755, 96syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
9895, 97mpbird 235 . . . . . . . . 9  |-  ( ph  ->  A  e.  S )
99 pgpfac1.mg . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
10099subgmulgcl 16781 . . . . . . . . 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 1264 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  S )
102 prmz 14597 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1035, 102syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  ZZ )
10410, 99mulgcl 16726 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
1058, 103, 71, 104syl3anc 1264 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
10610, 99mulgcl 16726 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  A  e.  B )  ->  ( M  .x.  A )  e.  B )
1078, 91, 56, 106syl3anc 1264 . . . . . . . . . . 11  |-  ( ph  ->  ( M  .x.  A
)  e.  B )
108 eqid 2429 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
10910, 99, 108mulgdi 17402 . . . . . . . . . . 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 1266 . . . . . . . . . 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 6320 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( E 
.x.  C ) )
11210, 99mulgass 16739 . . . . . . . . . . . . 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 1266 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11410, 11, 99, 65gexid 17168 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( E  .x.  C )  =  .0.  )
11571, 114syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( E  .x.  C
)  =  .0.  )
116111, 113, 1153eqtr3rd 2479 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11710, 99mulgass 16739 . . . . . . . . . . . 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 1266 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  ( ( E  /  P ) 
.x.  ( M  .x.  A ) ) )
119116, 118oveq12d 6323 . . . . . . . . . 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 16769 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  B
)
12160, 120syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  B )
122121, 101sseldd 3471 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )
12310, 108, 65grplid 16647 . . . . . . . . . . 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 665 . . . . . . . . . 10  |-  ( ph  ->  (  .0.  ( +g  `  G ) ( ( ( E  /  P
)  x.  M ) 
.x.  A ) )  =  ( ( ( E  /  P )  x.  M )  .x.  A ) )
125110, 119, 1243eqtr2d 2476 . . . . . . . . 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 16781 . . . . . . . . . 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 1264 . . . . . . . . 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 3656 . . . . . . 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 4027 . . . . . 6  |-  ( ( ( ( E  /  P )  x.  M
)  .x.  A )  e.  {  .0.  }  ->  ( ( ( E  /  P )  x.  M
)  .x.  A )  =  .0.  )
134132, 133syl 17 . . . . 5  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
135 pgpfac1.o . . . . . . 7  |-  O  =  ( od `  G
)
13610, 135, 99, 65oddvds 17138 . . . . . 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 1264 . . . . 5  |-  ( ph  ->  ( ( O `  A )  ||  (
( E  /  P
)  x.  M )  <-> 
( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
)
138134, 137mpbird 235 . . . 4  |-  ( ph  ->  ( O `  A
)  ||  ( ( E  /  P )  x.  M ) )
13986, 138eqbrtrrd 4448 . . 3  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  ||  ( ( E  /  P )  x.  M ) )
14089nnne0d 10654 . . . 4  |-  ( ph  ->  ( E  /  P
)  =/=  0 )
141 dvdscmulr 14309 . . . 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 1268 . . 3  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  ||  (
( E  /  P
)  x.  M )  <-> 
P  ||  M )
)
143139, 142mpbid 213 . 2  |-  ( ph  ->  P  ||  M )
14481, 143jca 534 1  |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783    \ cdif 3439    i^i cin 3441    C_ wss 3442    C. wpss 3443   (/)c0 3767   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   1oc1o 7183    ~~ cen 7574   Fincfn 7577   0cc0 9538   1c1 9539    x. cmul 9543    / cdiv 10268   NNcn 10609   NN0cn0 10869   ZZcz 10937   ^cexp 12269   #chash 12512    || cdvds 14283   Primecprime 14593    pCnt cpc 14749   Basecbs 15084   +g cplusg 15152   0gc0g 15297  Moorecmre 15439  mrClscmrc 15440  ACScacs 15442   Mndcmnd 16486   Grpcgrp 16620  .gcmg 16623  SubGrpcsubg 16762   odcod 17116  gExcgex 17117   pGrp cpgp 17118   LSSumclsm 17221   Abelcabl 17366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-acn 8375  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-eqg 16767  df-ga 16895  df-cntz 16922  df-od 17120  df-gex 17121  df-pgp 17122  df-lsm 17223  df-cmn 17367  df-abl 17368
This theorem is referenced by:  pgpfac1lem3  17645
  Copyright terms: Public domain W3C validator