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Theorem pgpfac1lem4 15329
Description: Lemma for pgpfac1 15331. (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 15326 . . . . . . 7  |-  ( ph  ->  ( P  .x.  C
)  e.  ( S 
.(+)  W ) )
21 ablgrp 15110 . . . . . . . . . . . 12  |-  ( G  e.  Abel  ->  G  e. 
Grp )
229, 21syl 15 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Grp )
233subgacs 14668 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
24 acsmre 13570 . . . . . . . . . . 11  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
2522, 23, 243syl 18 . . . . . . . . . 10  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
263subgss 14638 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2712, 26syl 15 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
2827, 13sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  A  e.  B )
291mrcsncl 13530 . . . . . . . . . 10  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
3025, 28, 29syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
312, 30syl5eqel 2380 . . . . . . . 8  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
327lsmcom 15166 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  =  ( W  .(+)  S ) )
339, 31, 14, 32syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( S  .(+)  W )  =  ( W  .(+)  S ) )
3420, 33eleqtrd 2372 . . . . . 6  |-  ( ph  ->  ( P  .x.  C
)  e.  ( W 
.(+)  S ) )
35 eqid 2296 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
3635, 7, 14, 31lsmelvalm 14978 . . . . . 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 201 . . . . 5  |-  ( ph  ->  E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w (
-g `  G )
s ) )
38 eqid 2296 . . . . . . . . . . 11  |-  ( k  e.  ZZ  |->  ( k 
.x.  A ) )  =  ( k  e.  ZZ  |->  ( k  .x.  A ) )
393, 19, 38, 1cycsubg2 14670 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  B )  ->  ( K `  { A } )  =  ran  ( k  e.  ZZ  |->  ( k  .x.  A
) ) )
4022, 28, 39syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( K `  { A } )  =  ran  ( k  e.  ZZ  |->  ( k  .x.  A
) ) )
412, 40syl5eq 2340 . . . . . . . 8  |-  ( ph  ->  S  =  ran  (
k  e.  ZZ  |->  ( k  .x.  A ) ) )
4241rexeqdv 2756 . . . . . . 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 5899 . . . . . . . . 9  |-  ( k 
.x.  A )  e. 
_V
4443rgenw 2623 . . . . . . . 8  |-  A. k  e.  ZZ  ( k  .x.  A )  e.  _V
45 oveq2 5882 . . . . . . . . . 10  |-  ( s  =  ( k  .x.  A )  ->  (
w ( -g `  G
) s )  =  ( w ( -g `  G ) ( k 
.x.  A ) ) )
4645eqeq2d 2307 . . . . . . . . 9  |-  ( s  =  ( k  .x.  A )  ->  (
( P  .x.  C
)  =  ( w ( -g `  G
) s )  <->  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
4738, 46rexrnmpt 5686 . . . . . . . 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 8 . . . . . . 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 252 . . . . . 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 2577 . . . . 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 201 . . . 4  |-  ( ph  ->  E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
52 rexcom 2714 . . . 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 188 . . 3  |-  ( ph  ->  E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
5422ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  G  e.  Grp )
553subgss 14638 . . . . . . . . . . 11  |-  ( W  e.  (SubGrp `  G
)  ->  W  C_  B
)
5614, 55syl 15 . . . . . . . . . 10  |-  ( ph  ->  W  C_  B )
5756adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ZZ )  ->  W  C_  B )
5857sselda 3193 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  w  e.  B )
59 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  k  e.  ZZ )
6028ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  A  e.  B )
613, 19mulgcl 14600 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  k  e.  ZZ  /\  A  e.  B )  ->  (
k  .x.  A )  e.  B )
6254, 59, 60, 61syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
k  .x.  A )  e.  B )
63 pgpprm 14920 . . . . . . . . . . 11  |-  ( P pGrp 
G  ->  P  e.  Prime )
64 prmz 12778 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ZZ )
658, 63, 643syl 18 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ZZ )
66 eldifi 3311 . . . . . . . . . . . 12  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
6718, 66syl 15 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  U )
6827, 67sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  C  e.  B )
693, 19mulgcl 14600 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
7022, 65, 68, 69syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
7170ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  ( P  .x.  C )  e.  B )
72 eqid 2296 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
733, 72, 35grpsubadd 14569 . . . . . . . 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 ) )
7454, 58, 62, 71, 73syl13anc 1184 . . . . . . 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 ) )
75 eqcom 2298 . . . . . . 7  |-  ( ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  ( w
( -g `  G ) ( k  .x.  A
) )  =  ( P  .x.  C ) )
76 eqcom 2298 . . . . . . 7  |-  ( w  =  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w )
7774, 75, 763bitr4g 279 . . . . . 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 ) ) ) )
7877rexbidva 2573 . . . . 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 ) ) ) )
79 risset 2603 . . . . 5  |-  ( ( ( P  .x.  C
) ( +g  `  G
) ( k  .x.  A ) )  e.  W  <->  E. w  e.  W  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) ) )
8078, 79syl6bbr 254 . . . 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
) )
8180rexbidva 2573 . . 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
) )
8253, 81mpbid 201 . 2  |-  ( ph  ->  E. k  e.  ZZ  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) )  e.  W )
838adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  P pGrp  G )
849adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  G  e.  Abel )
8510adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  B  e.  Fin )
8611adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( O `  A
)  =  E )
8712adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  U  e.  (SubGrp `  G
) )
8813adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  A  e.  U )
8914adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  W  e.  (SubGrp `  G
) )
9015adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( S  i^i  W
)  =  {  .0.  } )
9116adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( S  .(+)  W ) 
C_  U )
9217adantr 451 . . . . 5  |-  ( (
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 ) )
9318adantr 451 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )
94 simprl 732 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
k  e.  ZZ )
95 simprr 733 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) )  e.  W )
96 eqid 2296 . . . . 5  |-  ( C ( +g  `  G
) ( ( k  /  P )  .x.  A ) )  =  ( C ( +g  `  G ) ( ( k  /  P ) 
.x.  A ) )
971, 2, 3, 4, 5, 6, 7, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 19, 94, 95, 96pgpfac1lem3 15328 . . . 4  |-  ( (
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 ) )
9897expr 598 . . 3  |-  ( (
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 ) ) )
9998rexlimdva 2680 . 2  |-  ( ph  ->  ( E. 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 ) ) )
10082, 99mpd 14 1  |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165    C. wpss 3166   {csn 3653   class class class wbr 4039    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   Fincfn 6879    / cdiv 9439   ZZcz 10040   Primecprime 12774   Basecbs 13164   +g cplusg 13224   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378   -gcsg 14381  .gcmg 14382  SubGrpcsubg 14631   odcod 14856  gExcgex 14857   pGrp cpgp 14858   LSSumclsm 14961   Abelcabel 15106
This theorem is referenced by:  pgpfac1lem5  15330
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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