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Theorem pgpfac1lem3a 15327
Description: Lemma for pgpfac1 15331. (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 3312 . . . 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 14920 . . . . . . . 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 15110 . . . . . . . . 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 14916 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
149, 10, 13syl2anc 642 . . . . . . 7  |-  ( ph  ->  E  e.  NN )
15 pceq0 12939 . . . . . . 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 5882 . . . . . 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 14527 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  B  =/=  (/) )
209, 19syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  (/) )
21 hashnncl 11370 . . . . . . . . . . . . 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 12919 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( # `
 B ) )  e.  NN0 )
2511, 12gexdvds3 14917 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  ||  ( # `  B ) )
269, 10, 25syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  E  ||  ( # `  B ) )
2711pgphash 14934 . . . . . . . . . . . 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 4063 . . . . . . . . . 10  |-  ( ph  ->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) )
30 oveq2 5882 . . . . . . . . . . . 12  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( P ^ k )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
3130breq2d 4051 . . . . . . . . . . 11  |-  ( k  =  ( P  pCnt  (
# `  B )
)  ->  ( E  ||  ( P ^ k
)  <->  E  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
3231rspcev 2897 . . . . . . . . . 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 12950 . . . . . . . . . 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 2301 . . . . . . 7  |-  ( ph  ->  ( P ^ ( P  pCnt  E ) )  =  E )
38 prmnn 12777 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
396, 38syl 15 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
4039nncnd 9778 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
4140exp0d 11255 . . . . . . 7  |-  ( ph  ->  ( P ^ 0 )  =  1 )
4237, 41eqeq12d 2310 . . . . . 6  |-  ( ph  ->  ( ( P ^
( P  pCnt  E
) )  =  ( P ^ 0 )  <-> 
E  =  1 ) )
43 grpmnd 14510 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
449, 43syl 15 . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
4511, 12gex1 14918 . . . . . . 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 14668 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
519, 50syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubGrp `  G )  e.  (ACS `  B )
)
52 acsmre 13570 . . . . . . . . . . . . 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 14638 . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  B )
59 pgpfac1.k . . . . . . . . . . . . 13  |-  K  =  (mrCls `  (SubGrp `  G
) )
6059mrcsncl 13530 . . . . . . . . . . . 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 2380 . . . . . . . . . 10  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
63 pgpfac1.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
64 pgpfac1.l . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  G )
6564lsmsubg2 15167 . . . . . . . . . 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 14645 . . . . . . . . 9  |-  ( ( S  .(+)  W )  e.  (SubGrp `  G )  ->  .0.  e.  ( S 
.(+)  W ) )
6966, 68syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  e.  ( S 
.(+)  W ) )
7069snssd 3776 . . . . . . 7  |-  ( ph  ->  {  .0.  }  C_  ( S  .(+)  W ) )
7170adantr 451 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  {  .0.  } 
C_  ( S  .(+)  W ) )
72 eldifi 3311 . . . . . . . . . 10  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
731, 72syl 15 . . . . . . . . 9  |-  ( ph  ->  C  e.  U )
7456, 73sseldd 3194 . . . . . . . 8  |-  ( ph  ->  C  e.  B )
7574adantr 451 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e.  B )
7611, 67grpidcl 14526 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .0.  e.  B )
779, 76syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
78 en1eqsn 7104 . . . . . . . 8  |-  ( (  .0.  e.  B  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
7977, 78sylan 457 . . . . . . 7  |-  ( (
ph  /\  B  ~~  1o )  ->  B  =  {  .0.  } )
8075, 79eleqtrd 2372 . . . . . 6  |-  ( (
ph  /\  B  ~~  1o )  ->  C  e. 
{  .0.  } )
8171, 80sseldd 3194 . . . . 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 9778 . . . . . 6  |-  ( ph  ->  E  e.  CC )
8739nnne0d 9806 . . . . . 6  |-  ( ph  ->  P  =/=  0 )
8886, 40, 87divcan1d 9553 . . . . 5  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  =  E )
8985, 88eqtr4d 2331 . . . 4  |-  ( ph  ->  ( O `  A
)  =  ( ( E  /  P )  x.  P ) )
90 nndivdvds 12553 . . . . . . . . . . . . 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 10132 . . . . . . . . . 10  |-  ( ph  ->  ( E  /  P
)  e.  ZZ )
94 pgpfac1.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
9593, 94zmulcld 10139 . . . . . . . . 9  |-  ( ph  ->  ( ( E  /  P )  x.  M
)  e.  ZZ )
9658snssd 3776 . . . . . . . . . . . 12  |-  ( ph  ->  { A }  C_  B )
9759mrcssid 13535 . . . . . . . . . . . 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 3238 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  S )
100 snssg 3767 . . . . . . . . . . 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 14650 . . . . . . . . 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 12778 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1076, 106syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  ZZ )
10811, 103mulgcl 14600 . . . . . . . . . . . 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 14600 . . . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
11311, 103, 112mulgdi 15142 . . . . . . . . . . 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 5889 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( E  /  P )  x.  P )  .x.  C
)  =  ( E 
.x.  C ) )
11611, 103mulgass 14613 . . . . . . . . . . . . 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 14908 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( E  .x.  C )  =  .0.  )
11974, 118syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( E  .x.  C
)  =  .0.  )
120115, 117, 1193eqtr3rd 2337 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =  ( ( E  /  P ) 
.x.  ( P  .x.  C ) ) )
12111, 103mulgass 14613 . . . . . . . . . . . 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 5892 . . . . . . . . . 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 14638 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  B
)
12562, 124syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  B )
126125, 105sseldd 3194 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  B )
12711, 112, 67grplid 14528 . . . . . . . . . . 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 2334 . . . . . . . . 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 14650 . . . . . . . . . 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 2371 . . . . . . . 8  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  W )
134 elin 3371 . . . . . . . 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 2372 . . . . . 6  |-  ( ph  ->  ( ( ( E  /  P )  x.  M )  .x.  A
)  e.  {  .0.  } )
138 elsni 3677 . . . . . 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 14878 . . . . . 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 4061 . . 3  |-  ( ph  ->  ( ( E  /  P )  x.  P
)  ||  ( ( E  /  P )  x.  M ) )
14592nnne0d 9806 . . . 4  |-  ( ph  ->  ( E  /  P
)  =/=  0 )
146 dvdscmulr 12573 . . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    i^i cin 3164    C_ wss 3165    C. wpss 3166   (/)c0 3468   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   1oc1o 6488    ~~ cen 6876   Fincfn 6879   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   +g cplusg 13224   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Mndcmnd 14377   Grpcgrp 14378  .gcmg 14382  SubGrpcsubg 14631   odcod 14856  gExcgex 14857   pGrp cpgp 14858   LSSumclsm 14961   Abelcabel 15106
This theorem is referenced by:  pgpfac1lem3  15328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-ga 14760  df-cntz 14809  df-od 14860  df-gex 14861  df-pgp 14862  df-lsm 14963  df-cmn 15107  df-abl 15108
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