MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpfac1lem4 Structured version   Unicode version

Theorem pgpfac1lem4 17255
Description: Lemma for pgpfac1 17257. (Contributed by Mario Carneiro, 27-Apr-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 )
Assertion
Ref Expression
pgpfac1lem4  |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
Distinct variable groups:    t,  .0.    w, t, A    t,  .(+) , w   
t, P, w    t, B    t, G, w    t, U, w    t, C, w   
t, S, w    t, W, w    ph, t, w   
t,  .x. , w    t, K, w
Allowed substitution hints:    B( w)    E( w, t)    O( w, t)    .0. ( w)

Proof of Theorem pgpfac1lem4
Dummy variables  k 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pgpfac1.k . . . . . . . 8  |-  K  =  (mrCls `  (SubGrp `  G
) )
2 pgpfac1.s . . . . . . . 8  |-  S  =  ( K `  { A } )
3 pgpfac1.b . . . . . . . 8  |-  B  =  ( Base `  G
)
4 pgpfac1.o . . . . . . . 8  |-  O  =  ( od `  G
)
5 pgpfac1.e . . . . . . . 8  |-  E  =  (gEx `  G )
6 pgpfac1.z . . . . . . . 8  |-  .0.  =  ( 0g `  G )
7 pgpfac1.l . . . . . . . 8  |-  .(+)  =  (
LSSum `  G )
8 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
9 pgpfac1.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
10 pgpfac1.n . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
11 pgpfac1.oe . . . . . . . 8  |-  ( ph  ->  ( O `  A
)  =  E )
12 pgpfac1.u . . . . . . . 8  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
13 pgpfac1.au . . . . . . . 8  |-  ( ph  ->  A  e.  U )
14 pgpfac1.w . . . . . . . 8  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
15 pgpfac1.i . . . . . . . 8  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
16 pgpfac1.ss . . . . . . . 8  |-  ( ph  ->  ( S  .(+)  W ) 
C_  U )
17 pgpfac1.2 . . . . . . . 8  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w )
)
18 pgpfac1.c . . . . . . . 8  |-  ( ph  ->  C  e.  ( U 
\  ( S  .(+)  W ) ) )
19 pgpfac1.mg . . . . . . . 8  |-  .x.  =  (.g
`  G )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19pgpfac1lem2 17252 . . . . . . 7  |-  ( ph  ->  ( P  .x.  C
)  e.  ( S 
.(+)  W ) )
21 ablgrp 16929 . . . . . . . . . . . 12  |-  ( G  e.  Abel  ->  G  e. 
Grp )
229, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Grp )
233subgacs 16362 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
24 acsmre 15068 . . . . . . . . . . 11  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
2522, 23, 243syl 20 . . . . . . . . . 10  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
263subgss 16328 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2712, 26syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
2827, 13sseldd 3500 . . . . . . . . . 10  |-  ( ph  ->  A  e.  B )
291mrcsncl 15028 . . . . . . . . . 10  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
3025, 28, 29syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
312, 30syl5eqel 2549 . . . . . . . 8  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
327lsmcom 16990 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  =  ( W  .(+)  S ) )
339, 31, 14, 32syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( S  .(+)  W )  =  ( W  .(+)  S ) )
3420, 33eleqtrd 2547 . . . . . 6  |-  ( ph  ->  ( P  .x.  C
)  e.  ( W 
.(+)  S ) )
35 eqid 2457 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
3635, 7, 14, 31lsmelvalm 16797 . . . . . 6  |-  ( ph  ->  ( ( P  .x.  C )  e.  ( W  .(+)  S )  <->  E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G ) s ) ) )
3734, 36mpbid 210 . . . . 5  |-  ( ph  ->  E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w (
-g `  G )
s ) )
38 eqid 2457 . . . . . . . . . . 11  |-  ( k  e.  ZZ  |->  ( k 
.x.  A ) )  =  ( k  e.  ZZ  |->  ( k  .x.  A ) )
393, 19, 38, 1cycsubg2 16364 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  B )  ->  ( K `  { A } )  =  ran  ( k  e.  ZZ  |->  ( k  .x.  A
) ) )
4022, 28, 39syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( K `  { A } )  =  ran  ( k  e.  ZZ  |->  ( k  .x.  A
) ) )
412, 40syl5eq 2510 . . . . . . . 8  |-  ( ph  ->  S  =  ran  (
k  e.  ZZ  |->  ( k  .x.  A ) ) )
4241rexeqdv 3061 . . . . . . 7  |-  ( ph  ->  ( E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. s  e.  ran  ( k  e.  ZZ  |->  ( k  .x.  A ) ) ( P  .x.  C )  =  ( w (
-g `  G )
s ) ) )
43 ovex 6324 . . . . . . . . 9  |-  ( k 
.x.  A )  e. 
_V
4443rgenw 2818 . . . . . . . 8  |-  A. k  e.  ZZ  ( k  .x.  A )  e.  _V
45 oveq2 6304 . . . . . . . . . 10  |-  ( s  =  ( k  .x.  A )  ->  (
w ( -g `  G
) s )  =  ( w ( -g `  G ) ( k 
.x.  A ) ) )
4645eqeq2d 2471 . . . . . . . . 9  |-  ( s  =  ( k  .x.  A )  ->  (
( P  .x.  C
)  =  ( w ( -g `  G
) s )  <->  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
4738, 46rexrnmpt 6042 . . . . . . . 8  |-  ( A. k  e.  ZZ  (
k  .x.  A )  e.  _V  ->  ( E. s  e.  ran  ( k  e.  ZZ  |->  ( k 
.x.  A ) ) ( P  .x.  C
)  =  ( w ( -g `  G
) s )  <->  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
4844, 47ax-mp 5 . . . . . . 7  |-  ( E. s  e.  ran  (
k  e.  ZZ  |->  ( k  .x.  A ) ) ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) )
4942, 48syl6bb 261 . . . . . 6  |-  ( ph  ->  ( E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
5049rexbidv 2968 . . . . 5  |-  ( ph  ->  ( E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
5137, 50mpbid 210 . . . 4  |-  ( ph  ->  E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
52 rexcom 3019 . . . 4  |-  ( E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G ) ( k 
.x.  A ) )  <->  E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
5351, 52sylib 196 . . 3  |-  ( ph  ->  E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
5422ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  G  e.  Grp )
553subgss 16328 . . . . . . . . . . 11  |-  ( W  e.  (SubGrp `  G
)  ->  W  C_  B
)
5614, 55syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  C_  B )
5756adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ZZ )  ->  W  C_  B )
5857sselda 3499 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  w  e.  B )
59 simplr 755 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  k  e.  ZZ )
6028ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  A  e.  B )
613, 19mulgcl 16285 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  k  e.  ZZ  /\  A  e.  B )  ->  (
k  .x.  A )  e.  B )
6254, 59, 60, 61syl3anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
k  .x.  A )  e.  B )
63 pgpprm 16739 . . . . . . . . . . 11  |-  ( P pGrp 
G  ->  P  e.  Prime )
64 prmz 14232 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ZZ )
658, 63, 643syl 20 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ZZ )
6618eldifad 3483 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  U )
6727, 66sseldd 3500 . . . . . . . . . 10  |-  ( ph  ->  C  e.  B )
683, 19mulgcl 16285 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
6922, 65, 67, 68syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
7069ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  ( P  .x.  C )  e.  B )
71 eqid 2457 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
723, 71, 35grpsubadd 16252 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( w  e.  B  /\  ( k  .x.  A
)  e.  B  /\  ( P  .x.  C )  e.  B ) )  ->  ( ( w ( -g `  G
) ( k  .x.  A ) )  =  ( P  .x.  C
)  <->  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w ) )
7354, 58, 62, 70, 72syl13anc 1230 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
( w ( -g `  G ) ( k 
.x.  A ) )  =  ( P  .x.  C )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w ) )
74 eqcom 2466 . . . . . . 7  |-  ( ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  ( w
( -g `  G ) ( k  .x.  A
) )  =  ( P  .x.  C ) )
75 eqcom 2466 . . . . . . 7  |-  ( w  =  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w )
7673, 74, 753bitr4g 288 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
( P  .x.  C
)  =  ( w ( -g `  G
) ( k  .x.  A ) )  <->  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) ) ) )
7776rexbidva 2965 . . . . 5  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  E. w  e.  W  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) ) ) )
78 risset 2982 . . . . 5  |-  ( ( ( P  .x.  C
) ( +g  `  G
) ( k  .x.  A ) )  e.  W  <->  E. w  e.  W  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) ) )
7977, 78syl6bbr 263 . . . 4  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  e.  W
) )
8079rexbidva 2965 . . 3  |-  ( ph  ->  ( E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) )  <->  E. k  e.  ZZ  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  e.  W
) )
8153, 80mpbid 210 . 2  |-  ( ph  ->  E. k  e.  ZZ  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) )  e.  W )
828adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  P pGrp  G )
839adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  G  e.  Abel )
8410adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  B  e.  Fin )
8511adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( O `  A
)  =  E )
8612adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  U  e.  (SubGrp `  G
) )
8713adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  A  e.  U )
8814adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  W  e.  (SubGrp `  G
) )
8915adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( S  i^i  W
)  =  {  .0.  } )
9016adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( S  .(+)  W ) 
C_  U )
9117adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w )
)
9218adantr 465 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )
93 simprl 756 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
k  e.  ZZ )
94 simprr 757 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) )  e.  W )
95 eqid 2457 . . 3  |-  ( C ( +g  `  G
) ( ( k  /  P )  .x.  A ) )  =  ( C ( +g  `  G ) ( ( k  /  P ) 
.x.  A ) )
961, 2, 3, 4, 5, 6, 7, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 19, 93, 94, 95pgpfac1lem3 17254 . 2  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
9781, 96rexlimddv 2953 1  |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471    C. wpss 3472   {csn 4032   class class class wbr 4456    |-> cmpt 4515   ran crn 5009   ` cfv 5594  (class class class)co 6296   Fincfn 7535    / cdiv 10227   ZZcz 10885   Primecprime 14228   Basecbs 14643   +g cplusg 14711   0gc0g 14856  Moorecmre 14998  mrClscmrc 14999  ACScacs 15001   Grpcgrp 16179   -gcsg 16181  .gcmg 16182  SubGrpcsubg 16321   odcod 16675  gExcgex 16676   pGrp cpgp 16677   LSSumclsm 16780   Abelcabl 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-dvds 13998  df-gcd 14156  df-prm 14229  df-pc 14372  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-eqg 16326  df-ga 16454  df-cntz 16481  df-od 16679  df-gex 16680  df-pgp 16681  df-lsm 16782  df-cmn 16926  df-abl 16927
This theorem is referenced by:  pgpfac1lem5  17256
  Copyright terms: Public domain W3C validator