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Theorem lcfrlem23 31807
Description: Lemma for lcfr 31827. TODO: this proof was built from other proof pieces that may change  N `  { X ,  Y } into subspace sum and back unnecessarily, or similar things. (Contributed by NM, 1-Mar-2015.)
Hypotheses
Ref Expression
lcfrlem17.h  |-  H  =  ( LHyp `  K
)
lcfrlem17.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcfrlem17.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcfrlem17.v  |-  V  =  ( Base `  U
)
lcfrlem17.p  |-  .+  =  ( +g  `  U )
lcfrlem17.z  |-  .0.  =  ( 0g `  U )
lcfrlem17.n  |-  N  =  ( LSpan `  U )
lcfrlem17.a  |-  A  =  (LSAtoms `  U )
lcfrlem17.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfrlem17.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
lcfrlem17.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
lcfrlem17.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
lcfrlem22.b  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
lcfrlem23.s  |-  .(+)  =  (
LSSum `  U )
Assertion
Ref Expression
lcfrlem23  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
)  .(+)  B )  =  (  ._|_  `  { ( X  .+  Y ) } ) )

Proof of Theorem lcfrlem23
StepHypRef Expression
1 lcfrlem22.b . . . . . . 7  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
21fveq2i 5608 . . . . . 6  |-  (  ._|_  `  B )  =  ( 
._|_  `  ( ( N `
 { X ,  Y } )  i^i  (  ._|_  `  { ( X 
.+  Y ) } ) ) )
3 lcfrlem17.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
4 eqid 2358 . . . . . . . 8  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
5 lcfrlem17.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
6 lcfrlem17.v . . . . . . . 8  |-  V  =  ( Base `  U
)
7 lcfrlem17.o . . . . . . . 8  |-  ._|_  =  ( ( ocH `  K
) `  W )
8 eqid 2358 . . . . . . . 8  |-  ( (joinH `  K ) `  W
)  =  ( (joinH `  K ) `  W
)
9 lcfrlem17.k . . . . . . . 8  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 lcfrlem17.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
11 lcfrlem17.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
12 eldifi 3374 . . . . . . . . . 10  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1311, 12syl 15 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
14 lcfrlem17.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
15 eldifi 3374 . . . . . . . . . 10  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1614, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
173, 5, 6, 10, 4, 9, 13, 16dihprrn 31668 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
183, 5, 9dvhlmod 31352 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
19 lcfrlem17.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  U )
206, 19lmodvacl 15734 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
2118, 13, 16, 20syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
2221snssd 3839 . . . . . . . . 9  |-  ( ph  ->  { ( X  .+  Y ) }  C_  V )
233, 4, 5, 6, 7dochcl 31595 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { ( X 
.+  Y ) } 
C_  V )  -> 
(  ._|_  `  { ( X  .+  Y ) } )  e.  ran  (
( DIsoH `  K ) `  W ) )
249, 22, 23syl2anc 642 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
253, 4, 5, 6, 7, 8, 9, 17, 24dochdmm1 31652 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  { ( X 
.+  Y ) } ) ) ) )
263, 5, 7, 6, 10, 9, 21dochocsn 31623 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  { ( X  .+  Y ) } ) )  =  ( N `
 { ( X 
.+  Y ) } ) )
2726oveq2d 5958 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  ( N `  { X ,  Y } ) ) ( (joinH `  K
) `  W )
(  ._|_  `  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `
 { X ,  Y } ) ) ( (joinH `  K ) `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
28 lcfrlem23.s . . . . . . . 8  |-  .(+)  =  (
LSSum `  U )
29 prssi 3850 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  { X ,  Y }  C_  V )
3013, 16, 29syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  { X ,  Y }  C_  V )
316, 10lspssv 15833 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  { X ,  Y }  C_  V )  ->  ( N `  { X ,  Y } )  C_  V )
3218, 30, 31syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( N `  { X ,  Y }
)  C_  V )
333, 4, 5, 6, 7dochcl 31595 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  C_  V )  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
349, 32, 33syl2anc 642 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
353, 5, 6, 28, 10, 4, 8, 9, 34, 21dihjat1 31671 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  ( N `  { X ,  Y } ) ) ( (joinH `  K
) `  W )
( N `  {
( X  .+  Y
) } ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3625, 27, 353eqtrd 2394 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
372, 36syl5eq 2402 . . . . 5  |-  ( ph  ->  (  ._|_  `  B )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3837ineq2d 3446 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  i^i  ( (  ._|_  `  ( N `  { X ,  Y } ) ) 
.(+)  ( N `  { ( X  .+  Y ) } ) ) ) )
39 eqid 2358 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4039lsssssubg 15808 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
4118, 40syl 15 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
426, 39, 10lspsncl 15827 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
4318, 13, 42syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
446, 39, 10lspsncl 15827 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
4518, 16, 44syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
4639, 28lsmcl 15929 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( N `  { X } )  e.  (
LSubSp `  U )  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  e.  ( LSubSp `  U
) )
4718, 43, 45, 46syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  ( LSubSp `  U )
)
4841, 47sseldd 3257 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  (SubGrp `  U )
)
493, 5, 6, 39, 7dochlss 31596 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  C_  V )  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  ( LSubSp `  U )
)
509, 32, 49syl2anc 642 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  ( LSubSp `  U )
)
5141, 50sseldd 3257 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  (SubGrp `  U )
)
526, 39, 10lspsncl 15827 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  e.  ( LSubSp `  U ) )
5318, 21, 52syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  ( LSubSp `  U )
)
5441, 53sseldd 3257 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  U )
)
556, 19, 10, 28lspsntri 15943 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  C_  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
5618, 13, 16, 55syl3anc 1182 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } )  .(+)  ( N `  { Y } ) ) )
5728lsmmod2 15078 . . . . 5  |-  ( ( ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  e.  (SubGrp `  U
)  /\  (  ._|_  `  ( N `  { X ,  Y }
) )  e.  (SubGrp `  U )  /\  ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  U ) )  /\  ( N `  { ( X  .+  Y ) } )  C_  (
( N `  { X } )  .(+)  ( N `
 { Y }
) ) )  -> 
( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )  =  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) ) )
5848, 51, 54, 56, 57syl31anc 1185 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )  =  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) ) )
596, 10, 28, 18, 13, 16lsmpr 15935 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
6059ineq1d 3445 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  ( N `  { X ,  Y } ) ) ) )
616, 39, 10, 18, 13, 16lspprcl 15828 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
62 lcfrlem17.z . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
633, 5, 39, 62, 7dochnoncon 31633 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )  -> 
( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  {  .0.  } )
649, 61, 63syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  {  .0.  } )
6560, 64eqtr3d 2392 . . . . . 6  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  ( N `  { X ,  Y } ) ) )  =  {  .0.  } )
6665oveq1d 5957 . . . . 5  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) ) )
6762, 28lsm02 15074 . . . . . 6  |-  ( ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { ( X 
.+  Y ) } ) )  =  ( N `  { ( X  .+  Y ) } ) )
6854, 67syl 15 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) )  =  ( N `  { ( X  .+  Y ) } ) )
6966, 68eqtrd 2390 . . . 4  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( N `  {
( X  .+  Y
) } ) )
7038, 58, 693eqtrd 2394 . . 3  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( N `  { ( X  .+  Y ) } ) )
7170fveq2d 5609 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  (  ._|_  `  ( N `  {
( X  .+  Y
) } ) ) )
723, 5, 6, 28, 10, 4, 9, 13, 16dihsmsnrn 31677 . . . 4  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
73 lcfrlem17.a . . . . . 6  |-  A  =  (LSAtoms `  U )
74 lcfrlem17.ne . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
753, 7, 5, 6, 19, 62, 10, 73, 9, 11, 14, 74, 1lcfrlem22 31806 . . . . . 6  |-  ( ph  ->  B  e.  A )
766, 73, 18, 75lsatssv 29240 . . . . 5  |-  ( ph  ->  B  C_  V )
773, 4, 5, 6, 7dochcl 31595 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  C_  V
)  ->  (  ._|_  `  B )  e.  ran  ( ( DIsoH `  K
) `  W )
)
789, 76, 77syl2anc 642 . . . 4  |-  ( ph  ->  (  ._|_  `  B )  e.  ran  ( (
DIsoH `  K ) `  W ) )
793, 4, 5, 6, 7, 8, 9, 72, 78dochdmm1 31652 . . 3  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) ) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  B ) ) ) )
8059fveq2d 5609 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) )
813, 5, 7, 6, 10, 9, 30dochocsp 31621 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  { X ,  Y } ) )
8280, 81eqtr3d 2392 . . . 4  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )  =  (  ._|_  `  { X ,  Y } ) )
833, 5, 4, 73dih1dimat 31572 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  e.  A
)  ->  B  e.  ran  ( ( DIsoH `  K
) `  W )
)
849, 75, 83syl2anc 642 . . . . 5  |-  ( ph  ->  B  e.  ran  (
( DIsoH `  K ) `  W ) )
853, 4, 7dochoc 31609 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  B ) )  =  B )
869, 84, 85syl2anc 642 . . . 4  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  B ) )  =  B )
8782, 86oveq12d 5960 . . 3  |-  ( ph  ->  ( (  ._|_  `  (
( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) ( (joinH `  K ) `  W ) (  ._|_  `  (  ._|_  `  B ) ) )  =  ( (  ._|_  `  { X ,  Y } ) ( (joinH `  K ) `  W ) B ) )
883, 4, 5, 6, 7dochcl 31595 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { X ,  Y }  C_  V )  ->  (  ._|_  `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
899, 30, 88syl2anc 642 . . . 4  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
903, 4, 8, 5, 28, 73, 9, 89, 75dihjat2 31673 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( (joinH `  K ) `  W
) B )  =  ( (  ._|_  `  { X ,  Y }
)  .(+)  B ) )
9179, 87, 903eqtrd 2394 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  { X ,  Y } )  .(+)  B ) )
923, 5, 7, 6, 10, 9, 22dochocsp 31621 . 2  |-  ( ph  ->  (  ._|_  `  ( N `
 { ( X 
.+  Y ) } ) )  =  ( 
._|_  `  { ( X 
.+  Y ) } ) )
9371, 91, 923eqtr3d 2398 1  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
)  .(+)  B )  =  (  ._|_  `  { ( X  .+  Y ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225    i^i cin 3227    C_ wss 3228   {csn 3716   {cpr 3717   ran crn 4769   ` cfv 5334  (class class class)co 5942   Basecbs 13239   +g cplusg 13299   0gc0g 13493  SubGrpcsubg 14708   LSSumclsm 15038   LModclmod 15720   LSubSpclss 15782   LSpanclspn 15821  LSAtomsclsa 29216   HLchlt 29592   LHypclh 30225   DVecHcdvh 31320   DIsoHcdih 31470   ocHcoch 31589  joinHcdjh 31636
This theorem is referenced by:  lcfrlem25  31809  lcfrlem35  31819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-fal 1320  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-tpos 6318  df-undef 6382  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-sca 13315  df-vsca 13316  df-0g 13497  df-mre 13581  df-mrc 13582  df-acs 13584  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-p1 14239  df-lat 14245  df-clat 14307  df-mnd 14460  df-submnd 14509  df-grp 14582  df-minusg 14583  df-sbg 14584  df-subg 14711  df-cntz 14886  df-oppg 14912  df-lsm 15040  df-cmn 15184  df-abl 15185  df-mgp 15419  df-rng 15433  df-ur 15435  df-oppr 15498  df-dvdsr 15516  df-unit 15517  df-invr 15547  df-dvr 15558  df-drng 15607  df-lmod 15722  df-lss 15783  df-lsp 15822  df-lvec 15949  df-lsatoms 29218  df-lshyp 29219  df-lcv 29261  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-lvols 29741  df-lines 29742  df-psubsp 29744  df-pmap 29745  df-padd 30037  df-lhyp 30229  df-laut 30230  df-ldil 30345  df-ltrn 30346  df-trl 30400  df-tgrp 30984  df-tendo 30996  df-edring 30998  df-dveca 31244  df-disoa 31271  df-dvech 31321  df-dib 31381  df-dic 31415  df-dih 31471  df-doch 31590  df-djh 31637
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