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Theorem lcfrlem23 31755
Description: Lemma for lcfr 31775. 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 5528 . . . . . 6  |-  (  ._|_  `  B )  =  ( 
._|_  `  ( ( N `
 { X ,  Y } )  i^i  (  ._|_  `  { ( X 
.+  Y ) } ) ) )
3 lcfrlem17.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
4 eqid 2283 . . . . . . . 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 2283 . . . . . . . 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 3298 . . . . . . . . . 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 3298 . . . . . . . . . 10  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1614, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
173, 5, 6, 10, 4, 9, 13, 16dihprrn 31616 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
183, 5, 9dvhlmod 31300 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
19 lcfrlem17.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  U )
206, 19lmodvacl 15641 . . . . . . . . . . 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 3760 . . . . . . . . 9  |-  ( ph  ->  { ( X  .+  Y ) }  C_  V )
233, 4, 5, 6, 7dochcl 31543 . . . . . . . . 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 31600 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  { ( X 
.+  Y ) } ) ) ) )
263, 5, 7, 6, 10, 9, 21dochocsn 31571 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  { ( X  .+  Y ) } ) )  =  ( N `
 { ( X 
.+  Y ) } ) )
2726oveq2d 5874 . . . . . . 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 3771 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  { X ,  Y }  C_  V )
3013, 16, 29syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  { X ,  Y }  C_  V )
316, 10lspssv 15740 . . . . . . . . . 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 31543 . . . . . . . . 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 31619 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  ( N `  { X ,  Y } ) ) ( (joinH `  K
) `  W )
( N `  {
( X  .+  Y
) } ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3625, 27, 353eqtrd 2319 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
372, 36syl5eq 2327 . . . . 5  |-  ( ph  ->  (  ._|_  `  B )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3837ineq2d 3370 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  i^i  ( (  ._|_  `  ( N `  { X ,  Y } ) ) 
.(+)  ( N `  { ( X  .+  Y ) } ) ) ) )
39 eqid 2283 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4039lsssssubg 15715 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
4118, 40syl 15 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
426, 39, 10lspsncl 15734 . . . . . . . 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 15734 . . . . . . . 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 15836 . . . . . . 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 3181 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  (SubGrp `  U )
)
493, 5, 6, 39, 7dochlss 31544 . . . . . . 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 3181 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  (SubGrp `  U )
)
526, 39, 10lspsncl 15734 . . . . . . 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 3181 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  U )
)
556, 19, 10, 28lspsntri 15850 . . . . . 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 14985 . . . . 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 15842 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
6059ineq1d 3369 . . . . . . 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 15735 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
62 lcfrlem17.z . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
633, 5, 39, 62, 7dochnoncon 31581 . . . . . . . 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 2317 . . . . . 6  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  ( N `  { X ,  Y } ) ) )  =  {  .0.  } )
6665oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) ) )
6762, 28lsm02 14981 . . . . . 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 2315 . . . 4  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( N `  {
( X  .+  Y
) } ) )
7038, 58, 693eqtrd 2319 . . 3  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( N `  { ( X  .+  Y ) } ) )
7170fveq2d 5529 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  (  ._|_  `  ( N `  {
( X  .+  Y
) } ) ) )
723, 5, 6, 28, 10, 4, 9, 13, 16dihsmsnrn 31625 . . . 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 31754 . . . . . 6  |-  ( ph  ->  B  e.  A )
766, 73, 18, 75lsatssv 29188 . . . . 5  |-  ( ph  ->  B  C_  V )
773, 4, 5, 6, 7dochcl 31543 . . . . 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 31600 . . 3  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) ) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  B ) ) ) )
8059fveq2d 5529 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) )
813, 5, 7, 6, 10, 9, 30dochocsp 31569 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  { X ,  Y } ) )
8280, 81eqtr3d 2317 . . . 4  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )  =  (  ._|_  `  { X ,  Y } ) )
833, 5, 4, 73dih1dimat 31520 . . . . . 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 31557 . . . . 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 5876 . . 3  |-  ( ph  ->  ( (  ._|_  `  (
( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) ( (joinH `  K ) `  W ) (  ._|_  `  (  ._|_  `  B ) ) )  =  ( (  ._|_  `  { X ,  Y } ) ( (joinH `  K ) `  W ) B ) )
883, 4, 5, 6, 7dochcl 31543 . . . . 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 31621 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( (joinH `  K ) `  W
) B )  =  ( (  ._|_  `  { X ,  Y }
)  .(+)  B ) )
9179, 87, 903eqtrd 2319 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  { X ,  Y } )  .(+)  B ) )
923, 5, 7, 6, 10, 9, 22dochocsp 31569 . 2  |-  ( ph  ->  (  ._|_  `  ( N `
 { ( X 
.+  Y ) } ) )  =  ( 
._|_  `  { ( X 
.+  Y ) } ) )
9371, 91, 923eqtr3d 2323 1  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
)  .(+)  B )  =  (  ._|_  `  { ( X  .+  Y ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   {cpr 3641   ran crn 4690   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728  LSAtomsclsa 29164   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   DIsoHcdih 31418   ocHcoch 31537  joinHcdjh 31584
This theorem is referenced by:  lcfrlem25  31757  lcfrlem35  31767
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-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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585
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