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

Theorem pgpfac1lem3a 16999
Description: Lemma for pgpfac1 17003. (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 3494 . . 3  |-  ( ph  ->  -.  C  e.  ( S  .(+)  W )
)
3 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
4 pgpprm 16486 . . . . . . . 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 16676 . . . . . . . . 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 16482 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
138, 9, 12syl2anc 661 . . . . . . 7  |-  ( ph  ->  E  e.  NN )
14 pceq0 14270 . . . . . . 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 6303 . . . . . 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 15951 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  B  =/=  (/) )
198, 18syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  (/) )
20 hashnncl 12416 . . . . . . . . . . . . 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 14250 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 B ) )  e.  NN0 )
2410, 11gexdvds3 16483 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  ||  ( # `  B ) )
258, 9, 24syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  E  ||  ( # `  B ) )
2610pgphash 16500 . . . . . . . . . . . 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 4477 . . . . . . . . . 10  |-  ( ph  ->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) )
29 oveq2 6303 . . . . . . . . . . . 12  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( P ^ k )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
3029breq2d 4465 . . . . . . . . . . 11  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( E  ||  ( P ^ k
)  <->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
3130rspcev 3219 . . . . . . . . . 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 14281 . . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( P ^ ( P  pCnt  E ) )  =  E )
37 prmnn 14096 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
385, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
3938nncnd 10564 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
4039exp0d 12284 . . . . . . 7  |-  ( ph  ->  ( P ^ 0 )  =  1 )
4136, 40eqeq12d 2489 . . . . . 6  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
E  =  1 ) )
42 grpmnd 15934 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
438, 42syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
4410, 11gex1 16484 . . . . . . 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 16108 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
508, 49syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubGrp `  G )  e.  (ACS `  B )
)
5150acsmred 14928 . . . . . . . . . . . 12  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
52 pgpfac1.u . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5310subgss 16074 . . . . . . . . . . . . . 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 3510 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  B )
57 pgpfac1.k . . . . . . . . . . . . 13  |-  K  =  (mrCls `  (SubGrp `  G
) )
5857mrcsncl 14884 . . . . . . . . . . . 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 2559 . . . . . . . . . 10  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
61 pgpfac1.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
62 pgpfac1.l . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  G )
6362lsmsubg2 16738 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
646, 60, 61, 63syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
65 pgpfac1.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
6665subg0cl 16081 . . . . . . . . 9  |-  ( ( S  .(+)  W )  e.  (SubGrp `  G )  ->  .0.  e.  ( S 
.(+)  W ) )
6764, 66syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  ( S 
.(+)  W ) )
6867snssd 4178 . . . . . . 7  |-  ( ph  ->  {  .0.  }  C_  ( S  .(+)  W ) )
6968adantr 465 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  {  .0.  } 
C_  ( S  .(+)  W ) )
701eldifad 3493 . . . . . . . . 9  |-  ( ph  ->  C  e.  U )
7154, 70sseldd 3510 . . . . . . . 8  |-  ( ph  ->  C  e.  B )
7271adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  B )
7310, 65grpidcl 15950 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .0.  e.  B )
748, 73syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
75 en1eqsn 7761 . . . . . . . 8  |-  ( (  .0.  e.  B  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7674, 75sylan 471 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7772, 76eleqtrd 2557 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e. 
{  .0.  } )
7869, 77sseldd 3510 . . . . 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 10564 . . . . . 6  |-  ( ph  ->  E  e.  CC )
8438nnne0d 10592 . . . . . 6  |-  ( ph  ->  P  =/=  0 )
8583, 39, 84divcan1d 10333 . . . . 5  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  =  E )
8682, 85eqtr4d 2511 . . . 4  |-  ( ph  ->  ( O `  A
)  =  ( ( E  /  P )  x.  P ) )
87 nndivdvds 13870 . . . . . . . . . . . . 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 10977 . . . . . . . . . 10  |-  ( ph  ->  ( E  /  P
)  e.  ZZ )
91 pgpfac1.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
9290, 91zmulcld 10984 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  x.  M
)  e.  ZZ )
9356snssd 4178 . . . . . . . . . . . 12  |-  ( ph  ->  { A }  C_  B )
9451, 57, 93mrcssidd 14897 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
9594, 48syl6sseqr 3556 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  S )
96 snssg 4166 . . . . . . . . . . 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 16086 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  S )
102 prmz 14097 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1035, 102syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  ZZ )
10410, 99mulgcl 16031 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
1058, 103, 71, 104syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
10610, 99mulgcl 16031 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  A  e.  B )  ->  ( M  .x.  A )  e.  B )
1078, 91, 56, 106syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( M  .x.  A
)  e.  B )
108 eqid 2467 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
10910, 99, 108mulgdi 16708 . . . . . . . . . . 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 1230 . . . . . . . . . 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 6310 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( E 
.x.  C ) )
11210, 99mulgass 16044 . . . . . . . . . . . . 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 1230 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11410, 11, 99, 65gexid 16474 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( E  .x.  C )  =  .0.  )
11571, 114syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( E  .x.  C
)  =  .0.  )
116111, 113, 1153eqtr3rd 2517 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11710, 99mulgass 16044 . . . . . . . . . . . 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 1230 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  ( ( E  /  P ) 
.x.  ( M  .x.  A ) ) )
119116, 118oveq12d 6313 . . . . . . . . . 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 16074 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  B
)
12160, 120syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  B )
122121, 101sseldd 3510 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )
12310, 108, 65grplid 15952 . . . . . . . . . . 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 2514 . . . . . . . . 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 16086 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  e.  W )
129125, 128eqeltrrd 2556 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  W )
130101, 129elind 3693 . . . . . . 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 2557 . . . . . 6  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  {  .0.  } )
133 elsni 4058 . . . . . 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 16444 . . . . . 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 1228 . . . . 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 4475 . . 3  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  ||  ( ( E  /  P )  x.  M ) )
14089nnne0d 10592 . . . 4  |-  ( ph  ->  ( E  /  P
)  =/=  0 )
141 dvdscmulr 13890 . . . 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 1232 . . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    \ cdif 3478    i^i cin 3480    C_ wss 3481    C. wpss 3482   (/)c0 3790   {csn 4033   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   1oc1o 7135    ~~ cen 7525   Fincfn 7528   0cc0 9504   1c1 9505    x. cmul 9509    / cdiv 10218   NNcn 10548   NN0cn0 10807   ZZcz 10876   ^cexp 12146   #chash 12385    || cdivides 13864   Primecprime 14093    pCnt cpc 14236   Basecbs 14507   +g cplusg 14572   0gc0g 14712  Moorecmre 14854  mrClscmrc 14855  ACScacs 14857   Mndcmnd 15793   Grpcgrp 15925  .gcmg 15928  SubGrpcsubg 16067   odcod 16422  gExcgex 16423   pGrp cpgp 16424   LSSumclsm 16527   Abelcabl 16672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-eqg 16072  df-ga 16200  df-cntz 16227  df-od 16426  df-gex 16427  df-pgp 16428  df-lsm 16529  df-cmn 16673  df-abl 16674
This theorem is referenced by:  pgpfac1lem3  17000
  Copyright terms: Public domain W3C validator