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Theorem lcfrlem23 36763
Description: Lemma for lcfr 36783. 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 5875 . . . . . 6  |-  (  ._|_  `  B )  =  ( 
._|_  `  ( ( N `
 { X ,  Y } )  i^i  (  ._|_  `  { ( X 
.+  Y ) } ) ) )
3 lcfrlem17.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
4 eqid 2467 . . . . . . . 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 2467 . . . . . . . 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.  }
) )
1211eldifad 3493 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
13 lcfrlem17.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3493 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
153, 5, 6, 10, 4, 9, 12, 14dihprrn 36624 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
163, 5, 9dvhlmod 36308 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
17 lcfrlem17.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  U )
186, 17lmodvacl 17397 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
1916, 12, 14, 18syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
2019snssd 4178 . . . . . . . . 9  |-  ( ph  ->  { ( X  .+  Y ) }  C_  V )
213, 4, 5, 6, 7dochcl 36551 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { ( X 
.+  Y ) } 
C_  V )  -> 
(  ._|_  `  { ( X  .+  Y ) } )  e.  ran  (
( DIsoH `  K ) `  W ) )
229, 20, 21syl2anc 661 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
233, 4, 5, 6, 7, 8, 9, 15, 22dochdmm1 36608 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  { ( X 
.+  Y ) } ) ) ) )
243, 5, 7, 6, 10, 9, 19dochocsn 36579 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  { ( X  .+  Y ) } ) )  =  ( N `
 { ( X 
.+  Y ) } ) )
2524oveq2d 6311 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  ( N `  { X ,  Y } ) ) ( (joinH `  K
) `  W )
(  ._|_  `  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `
 { X ,  Y } ) ) ( (joinH `  K ) `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
26 lcfrlem23.s . . . . . . . 8  |-  .(+)  =  (
LSSum `  U )
27 prssi 4189 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  { X ,  Y }  C_  V )
2812, 14, 27syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  { X ,  Y }  C_  V )
296, 10lspssv 17500 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  { X ,  Y }  C_  V )  ->  ( N `  { X ,  Y } )  C_  V )
3016, 28, 29syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( N `  { X ,  Y }
)  C_  V )
313, 4, 5, 6, 7dochcl 36551 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  C_  V )  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
329, 30, 31syl2anc 661 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
333, 5, 6, 26, 10, 4, 8, 9, 32, 19dihjat1 36627 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  ( N `  { X ,  Y } ) ) ( (joinH `  K
) `  W )
( N `  {
( X  .+  Y
) } ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3423, 25, 333eqtrd 2512 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
352, 34syl5eq 2520 . . . . 5  |-  ( ph  ->  (  ._|_  `  B )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3635ineq2d 3705 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  i^i  ( (  ._|_  `  ( N `  { X ,  Y } ) ) 
.(+)  ( N `  { ( X  .+  Y ) } ) ) ) )
37 eqid 2467 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
3837lsssssubg 17475 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
3916, 38syl 16 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
406, 37, 10lspsncl 17494 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
4116, 12, 40syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
426, 37, 10lspsncl 17494 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
4316, 14, 42syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
4437, 26lsmcl 17600 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( N `  { X } )  e.  (
LSubSp `  U )  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  e.  ( LSubSp `  U
) )
4516, 41, 43, 44syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  ( LSubSp `  U )
)
4639, 45sseldd 3510 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  (SubGrp `  U )
)
473, 5, 6, 37, 7dochlss 36552 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  C_  V )  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  ( LSubSp `  U )
)
489, 30, 47syl2anc 661 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  ( LSubSp `  U )
)
4939, 48sseldd 3510 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  (SubGrp `  U )
)
506, 37, 10lspsncl 17494 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  e.  ( LSubSp `  U ) )
5116, 19, 50syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  ( LSubSp `  U )
)
5239, 51sseldd 3510 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  U )
)
536, 17, 10, 26lspsntri 17614 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  C_  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
5416, 12, 14, 53syl3anc 1228 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } )  .(+)  ( N `  { Y } ) ) )
5526lsmmod2 16567 . . . . 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 ) } ) ) )
5646, 49, 52, 54, 55syl31anc 1231 . . . 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 ) } ) ) )
576, 10, 26, 16, 12, 14lsmpr 17606 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
5857ineq1d 3704 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  ( N `  { X ,  Y } ) ) ) )
596, 37, 10, 16, 12, 14lspprcl 17495 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
60 lcfrlem17.z . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
613, 5, 37, 60, 7dochnoncon 36589 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )  -> 
( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  {  .0.  } )
629, 59, 61syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  {  .0.  } )
6358, 62eqtr3d 2510 . . . . . 6  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  ( N `  { X ,  Y } ) ) )  =  {  .0.  } )
6463oveq1d 6310 . . . . 5  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) ) )
6560, 26lsm02 16563 . . . . . 6  |-  ( ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { ( X 
.+  Y ) } ) )  =  ( N `  { ( X  .+  Y ) } ) )
6652, 65syl 16 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) )  =  ( N `  { ( X  .+  Y ) } ) )
6764, 66eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( N `  {
( X  .+  Y
) } ) )
6836, 56, 673eqtrd 2512 . . 3  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( N `  { ( X  .+  Y ) } ) )
6968fveq2d 5876 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  (  ._|_  `  ( N `  {
( X  .+  Y
) } ) ) )
703, 5, 6, 26, 10, 4, 9, 12, 14dihsmsnrn 36633 . . . 4  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
71 lcfrlem17.a . . . . . 6  |-  A  =  (LSAtoms `  U )
72 lcfrlem17.ne . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
733, 7, 5, 6, 17, 60, 10, 71, 9, 11, 13, 72, 1lcfrlem22 36762 . . . . . 6  |-  ( ph  ->  B  e.  A )
746, 71, 16, 73lsatssv 34196 . . . . 5  |-  ( ph  ->  B  C_  V )
753, 4, 5, 6, 7dochcl 36551 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  C_  V
)  ->  (  ._|_  `  B )  e.  ran  ( ( DIsoH `  K
) `  W )
)
769, 74, 75syl2anc 661 . . . 4  |-  ( ph  ->  (  ._|_  `  B )  e.  ran  ( (
DIsoH `  K ) `  W ) )
773, 4, 5, 6, 7, 8, 9, 70, 76dochdmm1 36608 . . 3  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) ) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  B ) ) ) )
7857fveq2d 5876 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) )
793, 5, 7, 6, 10, 9, 28dochocsp 36577 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  { X ,  Y } ) )
8078, 79eqtr3d 2510 . . . 4  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )  =  (  ._|_  `  { X ,  Y } ) )
813, 5, 4, 71dih1dimat 36528 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  e.  A
)  ->  B  e.  ran  ( ( DIsoH `  K
) `  W )
)
829, 73, 81syl2anc 661 . . . . 5  |-  ( ph  ->  B  e.  ran  (
( DIsoH `  K ) `  W ) )
833, 4, 7dochoc 36565 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  B ) )  =  B )
849, 82, 83syl2anc 661 . . . 4  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  B ) )  =  B )
8580, 84oveq12d 6313 . . 3  |-  ( ph  ->  ( (  ._|_  `  (
( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) ( (joinH `  K ) `  W ) (  ._|_  `  (  ._|_  `  B ) ) )  =  ( (  ._|_  `  { X ,  Y } ) ( (joinH `  K ) `  W ) B ) )
863, 4, 5, 6, 7dochcl 36551 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { X ,  Y }  C_  V )  ->  (  ._|_  `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
879, 28, 86syl2anc 661 . . . 4  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
883, 4, 8, 5, 26, 71, 9, 87, 73dihjat2 36629 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( (joinH `  K ) `  W
) B )  =  ( (  ._|_  `  { X ,  Y }
)  .(+)  B ) )
8977, 85, 883eqtrd 2512 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  { X ,  Y } )  .(+)  B ) )
903, 5, 7, 6, 10, 9, 20dochocsp 36577 . 2  |-  ( ph  ->  (  ._|_  `  ( N `
 { ( X 
.+  Y ) } ) )  =  ( 
._|_  `  { ( X 
.+  Y ) } ) )
9169, 89, 903eqtr3d 2516 1  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
)  .(+)  B )  =  (  ._|_  `  { ( X  .+  Y ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4033   {cpr 4035   ran crn 5006   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712  SubGrpcsubg 16067   LSSumclsm 16527   LModclmod 17383   LSubSpclss 17449   LSpanclspn 17488  LSAtomsclsa 34172   HLchlt 34548   LHypclh 35181   DVecHcdvh 36276   DIsoHcdih 36426   ocHcoch 36545  joinHcdjh 36592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-mre 14858  df-mrc 14859  df-acs 14861  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-oppg 16253  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lsatoms 34174  df-lshyp 34175  df-lcv 34217  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tgrp 35940  df-tendo 35952  df-edring 35954  df-dveca 36200  df-disoa 36227  df-dvech 36277  df-dib 36337  df-dic 36371  df-dih 36427  df-doch 36546  df-djh 36593
This theorem is referenced by:  lcfrlem25  36765  lcfrlem35  36775
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