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

Theorem pgpfac1lem3a 15311
Description: Lemma for pgpfac1 15315. (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 ) ) )
2 eldifn 3299 . . . 4  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  -.  C  e.  ( S  .(+)  W ) )
31, 2syl 15 . . 3  |-  ( ph  ->  -.  C  e.  ( S  .(+)  W )
)
4 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
5 pgpprm 14904 . . . . . . . 8  |-  ( P pGrp 
G  ->  P  e.  Prime )
64, 5syl 15 . . . . . . 7  |-  ( ph  ->  P  e.  Prime )
7 pgpfac1.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
8 ablgrp 15094 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
97, 8syl 15 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
10 pgpfac1.n . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
11 pgpfac1.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
12 pgpfac1.e . . . . . . . . 9  |-  E  =  (gEx `  G )
1311, 12gexcl2 14900 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
149, 10, 13syl2anc 642 . . . . . . 7  |-  ( ph  ->  E  e.  NN )
15 pceq0 12923 . . . . . . 7  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  (
( P  pCnt  E
)  =  0  <->  -.  P  ||  E ) )
166, 14, 15syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  E )  =  0  <->  -.  P  ||  E ) )
17 oveq2 5866 . . . . . 6  |-  ( ( P  pCnt  E )  =  0  ->  ( P ^ ( P  pCnt  E ) )  =  ( P ^ 0 ) )
1816, 17syl6bir 220 . . . . 5  |-  ( ph  ->  ( -.  P  ||  E  ->  ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 ) ) )
1911grpbn0 14511 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  B  =/=  (/) )
209, 19syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  (/) )
21 hashnncl 11354 . . . . . . . . . . . . 13  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
2210, 21syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
2320, 22mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  e.  NN )
246, 23pccld 12903 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 B ) )  e.  NN0 )
2511, 12gexdvds3 14901 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  ||  ( # `  B ) )
269, 10, 25syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  E  ||  ( # `  B ) )
2711pgphash 14918 . . . . . . . . . . . 12  |-  ( ( P pGrp  G  /\  B  e.  Fin )  ->  ( # `
 B )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
284, 10, 27syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
2926, 28breqtrd 4047 . . . . . . . . . 10  |-  ( ph  ->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) )
30 oveq2 5866 . . . . . . . . . . . 12  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( P ^ k )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
3130breq2d 4035 . . . . . . . . . . 11  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( E  ||  ( P ^ k
)  <->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
3231rspcev 2884 . . . . . . . . . 10  |-  ( ( ( P  pCnt  ( # `
 B ) )  e.  NN0  /\  E  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) )  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
3324, 29, 32syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  E. k  e.  NN0  E 
||  ( P ^
k ) )
34 pcprmpw2 12934 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  E  e.  NN )  ->  ( E. k  e.  NN0  E 
||  ( P ^
k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
356, 14, 34syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( E. k  e. 
NN0  E  ||  ( P ^ k )  <->  E  =  ( P ^ ( P 
pCnt  E ) ) ) )
3633, 35mpbid 201 . . . . . . . 8  |-  ( ph  ->  E  =  ( P ^ ( P  pCnt  E ) ) )
3736eqcomd 2288 . . . . . . 7  |-  ( ph  ->  ( P ^ ( P  pCnt  E ) )  =  E )
38 prmnn 12761 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
396, 38syl 15 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
4039nncnd 9762 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
4140exp0d 11239 . . . . . . 7  |-  ( ph  ->  ( P ^ 0 )  =  1 )
4237, 41eqeq12d 2297 . . . . . 6  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
E  =  1 ) )
43 grpmnd 14494 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
449, 43syl 15 . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
4511, 12gex1 14902 . . . . . . 7  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  B  ~~  1o ) )
4644, 45syl 15 . . . . . 6  |-  ( ph  ->  ( E  =  1  <-> 
B  ~~  1o )
)
4742, 46bitrd 244 . . . . 5  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
B  ~~  1o )
)
4818, 47sylibd 205 . . . 4  |-  ( ph  ->  ( -.  P  ||  E  ->  B  ~~  1o ) )
49 pgpfac1.s . . . . . . . . . . 11  |-  S  =  ( K `  { A } )
5011subgacs 14652 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
519, 50syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubGrp `  G )  e.  (ACS `  B )
)
52 acsmre 13554 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
5351, 52syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
54 pgpfac1.u . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5511subgss 14622 . . . . . . . . . . . . . 14  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
5654, 55syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  U  C_  B )
57 pgpfac1.au . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  U )
5856, 57sseldd 3181 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  B )
59 pgpfac1.k . . . . . . . . . . . . 13  |-  K  =  (mrCls `  (SubGrp `  G
) )
6059mrcsncl 13514 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
6153, 58, 60syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
6249, 61syl5eqel 2367 . . . . . . . . . 10  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
63 pgpfac1.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
64 pgpfac1.l . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  G )
6564lsmsubg2 15151 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
667, 62, 63, 65syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
67 pgpfac1.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
6867subg0cl 14629 . . . . . . . . 9  |-  ( ( S  .(+)  W )  e.  (SubGrp `  G )  ->  .0.  e.  ( S 
.(+)  W ) )
6966, 68syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  e.  ( S 
.(+)  W ) )
7069snssd 3760 . . . . . . 7  |-  ( ph  ->  {  .0.  }  C_  ( S  .(+)  W ) )
7170adantr 451 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  {  .0.  } 
C_  ( S  .(+)  W ) )
72 eldifi 3298 . . . . . . . . . 10  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
731, 72syl 15 . . . . . . . . 9  |-  ( ph  ->  C  e.  U )
7456, 73sseldd 3181 . . . . . . . 8  |-  ( ph  ->  C  e.  B )
7574adantr 451 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  B )
7611, 67grpidcl 14510 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .0.  e.  B )
779, 76syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
78 en1eqsn 7088 . . . . . . . 8  |-  ( (  .0.  e.  B  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7977, 78sylan 457 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
8075, 79eleqtrd 2359 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e. 
{  .0.  } )
8171, 80sseldd 3181 . . . . 5  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  ( S  .(+)  W ) )
8281ex 423 . . . 4  |-  ( ph  ->  ( B  ~~  1o  ->  C  e.  ( S 
.(+)  W ) ) )
8348, 82syld 40 . . 3  |-  ( ph  ->  ( -.  P  ||  E  ->  C  e.  ( S  .(+)  W )
) )
843, 83mt3d 117 . 2  |-  ( ph  ->  P  ||  E )
85 pgpfac1.oe . . . . 5  |-  ( ph  ->  ( O `  A
)  =  E )
8614nncnd 9762 . . . . . 6  |-  ( ph  ->  E  e.  CC )
8739nnne0d 9790 . . . . . 6  |-  ( ph  ->  P  =/=  0 )
8886, 40, 87divcan1d 9537 . . . . 5  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  =  E )
8985, 88eqtr4d 2318 . . . 4  |-  ( ph  ->  ( O `  A
)  =  ( ( E  /  P )  x.  P ) )
90 nndivdvds 12537 . . . . . . . . . . . . 13  |-  ( ( E  e.  NN  /\  P  e.  NN )  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
9114, 39, 90syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  ||  E  <->  ( E  /  P )  e.  NN ) )
9284, 91mpbid 201 . . . . . . . . . . 11  |-  ( ph  ->  ( E  /  P
)  e.  NN )
9392nnzd 10116 . . . . . . . . . 10  |-  ( ph  ->  ( E  /  P
)  e.  ZZ )
94 pgpfac1.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
9593, 94zmulcld 10123 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  x.  M
)  e.  ZZ )
9658snssd 3760 . . . . . . . . . . . 12  |-  ( ph  ->  { A }  C_  B )
9759mrcssid 13519 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { A }  C_  B )  ->  { A }  C_  ( K `  { A } ) )
9853, 96, 97syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
9998, 49syl6sseqr 3225 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  S )
100 snssg 3754 . . . . . . . . . . 11  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
10157, 100syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
10299, 101mpbird 223 . . . . . . . . 9  |-  ( ph  ->  A  e.  S )
103 pgpfac1.mg . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
104103subgmulgcl 14634 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  (
( E  /  P
)  x.  M )  e.  ZZ  /\  A  e.  S )  ->  (
( ( E  /  P )  x.  M
)  .x.  A )  e.  S )
10562, 95, 102, 104syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  S )
106 prmz 12762 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1076, 106syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  ZZ )
10811, 103mulgcl 14584 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
1099, 107, 74, 108syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
11011, 103mulgcl 14584 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  A  e.  B )  ->  ( M  .x.  A )  e.  B )
1119, 94, 58, 110syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( M  .x.  A
)  e.  B )
112 eqid 2283 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
11311, 103, 112mulgdi 15126 . . . . . . . . . . 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 ) ) ) )
1147, 93, 109, 111, 113syl13anc 1184 . . . . . . . . . 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 ) ) ) )
11588oveq1d 5873 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( E 
.x.  C ) )
11611, 103mulgass 14597 . . . . . . . . . . . . 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 ) ) )
1179, 93, 107, 74, 116syl13anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
11811, 12, 103, 67gexid 14892 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( E  .x.  C )  =  .0.  )
11974, 118syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( E  .x.  C
)  =  .0.  )
120115, 117, 1193eqtr3rd 2324 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
12111, 103mulgass 14597 . . . . . . . . . . . 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 ) ) )
1229, 93, 94, 58, 121syl13anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  ( ( E  /  P ) 
.x.  ( M  .x.  A ) ) )
123120, 122oveq12d 5876 . . . . . . . . . 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 ) ) ) )
12411subgss 14622 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  B
)
12562, 124syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  B )
126125, 105sseldd 3181 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )
12711, 112, 67grplid 14512 . . . . . . . . . . 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 ) )
1289, 126, 127syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  (  .0.  ( +g  `  G ) ( ( ( E  /  P
)  x.  M ) 
.x.  A ) )  =  ( ( ( E  /  P )  x.  M )  .x.  A ) )
129114, 123, 1283eqtr2d 2321 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  =  ( ( ( E  /  P )  x.  M )  .x.  A ) )
130 pgpfac1.mw . . . . . . . . . 10  |-  ( ph  ->  ( ( P  .x.  C ) ( +g  `  G ) ( M 
.x.  A ) )  e.  W )
131103subgmulgcl 14634 . . . . . . . . . 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 )
13263, 93, 130, 131syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  .x.  (
( P  .x.  C
) ( +g  `  G
) ( M  .x.  A ) ) )  e.  W )
133129, 132eqeltrrd 2358 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  W )
134 elin 3358 . . . . . . . 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 ) )
135105, 133, 134sylanbrc 645 . . . . . . 7  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  ( S  i^i  W ) )
136 pgpfac1.i . . . . . . 7  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
137135, 136eleqtrd 2359 . . . . . 6  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  {  .0.  } )
138 elsni 3664 . . . . . 6  |-  ( ( ( ( E  /  P )  x.  M
)  .x.  A )  e.  {  .0.  }  ->  ( ( ( E  /  P )  x.  M
)  .x.  A )  =  .0.  )
139137, 138syl 15 . . . . 5  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
140 pgpfac1.o . . . . . . 7  |-  O  =  ( od `  G
)
14111, 140, 103, 67oddvds 14862 . . . . . 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.  ) )
1429, 58, 95, 141syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( O `  A )  ||  (
( E  /  P
)  x.  M )  <-> 
( ( ( E  /  P )  x.  M )  .x.  A
)  =  .0.  )
)
143139, 142mpbird 223 . . . 4  |-  ( ph  ->  ( O `  A
)  ||  ( ( E  /  P )  x.  M ) )
14489, 143eqbrtrrd 4045 . . 3  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  ||  ( ( E  /  P )  x.  M ) )
14592nnne0d 9790 . . . 4  |-  ( ph  ->  ( E  /  P
)  =/=  0 )
146 dvdscmulr 12557 . . . 4  |-  ( ( P  e.  ZZ  /\  M  e.  ZZ  /\  (
( E  /  P
)  e.  ZZ  /\  ( E  /  P
)  =/=  0 ) )  ->  ( (
( E  /  P
)  x.  P ) 
||  ( ( E  /  P )  x.  M )  <->  P  ||  M
) )
147107, 94, 93, 145, 146syl112anc 1186 . . 3  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  ||  (
( E  /  P
)  x.  M )  <-> 
P  ||  M )
)
148144, 147mpbid 201 . 2  |-  ( ph  ->  P  ||  M )
14984, 148jca 518 1  |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ~~ cen 6860   Fincfn 6863   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104   #chash 11337    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   +g cplusg 13208   0gc0g 13400  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Mndcmnd 14361   Grpcgrp 14362  .gcmg 14366  SubGrpcsubg 14615   odcod 14840  gExcgex 14841   pGrp cpgp 14842   LSSumclsm 14945   Abelcabel 15090
This theorem is referenced by:  pgpfac1lem3  15312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ga 14744  df-cntz 14793  df-od 14844  df-gex 14845  df-pgp 14846  df-lsm 14947  df-cmn 15091  df-abl 15092
  Copyright terms: Public domain W3C validator