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Theorem lcfrlem23 35179
Description: Lemma for lcfr 35199. 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 5895 . . . . . 6  |-  (  ._|_  `  B )  =  ( 
._|_  `  ( ( N `
 { X ,  Y } )  i^i  (  ._|_  `  { ( X 
.+  Y ) } ) ) )
3 lcfrlem17.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
4 eqid 2462 . . . . . . . 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 2462 . . . . . . . 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 3428 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
13 lcfrlem17.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3428 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
153, 5, 6, 10, 4, 9, 12, 14dihprrn 35040 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
163, 5, 9dvhlmod 34724 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
17 lcfrlem17.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  U )
186, 17lmodvacl 18160 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
1916, 12, 14, 18syl3anc 1276 . . . . . . . . . 10  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
2019snssd 4130 . . . . . . . . 9  |-  ( ph  ->  { ( X  .+  Y ) }  C_  V )
213, 4, 5, 6, 7dochcl 34967 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { ( X 
.+  Y ) } 
C_  V )  -> 
(  ._|_  `  { ( X  .+  Y ) } )  e.  ran  (
( DIsoH `  K ) `  W ) )
229, 20, 21syl2anc 671 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
233, 4, 5, 6, 7, 8, 9, 15, 22dochdmm1 35024 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  { ( X 
.+  Y ) } ) ) ) )
243, 5, 7, 6, 10, 9, 19dochocsn 34995 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  { ( X  .+  Y ) } ) )  =  ( N `
 { ( X 
.+  Y ) } ) )
2524oveq2d 6336 . . . . . . 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 4141 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  { X ,  Y }  C_  V )
2812, 14, 27syl2anc 671 . . . . . . . . . 10  |-  ( ph  ->  { X ,  Y }  C_  V )
296, 10lspssv 18261 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  { X ,  Y }  C_  V )  ->  ( N `  { X ,  Y } )  C_  V )
3016, 28, 29syl2anc 671 . . . . . . . . 9  |-  ( ph  ->  ( N `  { X ,  Y }
)  C_  V )
313, 4, 5, 6, 7dochcl 34967 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  C_  V )  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
329, 30, 31syl2anc 671 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
333, 5, 6, 26, 10, 4, 8, 9, 32, 19dihjat1 35043 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  ( N `  { X ,  Y } ) ) ( (joinH `  K
) `  W )
( N `  {
( X  .+  Y
) } ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3423, 25, 333eqtrd 2500 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) ) )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
352, 34syl5eq 2508 . . . . 5  |-  ( ph  ->  (  ._|_  `  B )  =  ( (  ._|_  `  ( N `  { X ,  Y }
) )  .(+)  ( N `
 { ( X 
.+  Y ) } ) ) )
3635ineq2d 3646 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  i^i  ( (  ._|_  `  ( N `  { X ,  Y } ) ) 
.(+)  ( N `  { ( X  .+  Y ) } ) ) ) )
37 eqid 2462 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
3837lsssssubg 18236 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
3916, 38syl 17 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
406, 37, 10lspsncl 18255 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
4116, 12, 40syl2anc 671 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
426, 37, 10lspsncl 18255 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
4316, 14, 42syl2anc 671 . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
4437, 26lsmcl 18361 . . . . . . 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 1276 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  ( LSubSp `  U )
)
4639, 45sseldd 3445 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  e.  (SubGrp `  U )
)
473, 5, 6, 37, 7dochlss 34968 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X ,  Y }
)  C_  V )  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  ( LSubSp `  U )
)
489, 30, 47syl2anc 671 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  ( LSubSp `  U )
)
4939, 48sseldd 3445 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  e.  (SubGrp `  U )
)
506, 37, 10lspsncl 18255 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  e.  ( LSubSp `  U ) )
5116, 19, 50syl2anc 671 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  ( LSubSp `  U )
)
5239, 51sseldd 3445 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  U )
)
536, 17, 10, 26lspsntri 18375 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  C_  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
5416, 12, 14, 53syl3anc 1276 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } )  .(+)  ( N `  { Y } ) ) )
5526lsmmod2 17381 . . . . 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 1279 . . . 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 18367 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )
5857ineq1d 3645 . . . . . . 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 18256 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
60 lcfrlem17.z . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
613, 5, 37, 60, 7dochnoncon 35005 . . . . . . . 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 671 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  =  {  .0.  } )
6358, 62eqtr3d 2498 . . . . . 6  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  ( N `  { X ,  Y } ) ) )  =  {  .0.  } )
6463oveq1d 6335 . . . . 5  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) ) )
6560, 26lsm02 17377 . . . . . 6  |-  ( ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { ( X 
.+  Y ) } ) )  =  ( N `  { ( X  .+  Y ) } ) )
6652, 65syl 17 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  {
( X  .+  Y
) } ) )  =  ( N `  { ( X  .+  Y ) } ) )
6764, 66eqtrd 2496 . . . 4  |-  ( ph  ->  ( ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  ( N `  { X ,  Y }
) ) )  .(+)  ( N `  { ( X  .+  Y ) } ) )  =  ( N `  {
( X  .+  Y
) } ) )
6836, 56, 673eqtrd 2500 . . 3  |-  ( ph  ->  ( ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  B
) )  =  ( N `  { ( X  .+  Y ) } ) )
6968fveq2d 5896 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  (  ._|_  `  ( N `  {
( X  .+  Y
) } ) ) )
703, 5, 6, 26, 10, 4, 9, 12, 14dihsmsnrn 35049 . . . 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 35178 . . . . . 6  |-  ( ph  ->  B  e.  A )
746, 71, 16, 73lsatssv 32610 . . . . 5  |-  ( ph  ->  B  C_  V )
753, 4, 5, 6, 7dochcl 34967 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  C_  V
)  ->  (  ._|_  `  B )  e.  ran  ( ( DIsoH `  K
) `  W )
)
769, 74, 75syl2anc 671 . . . 4  |-  ( ph  ->  (  ._|_  `  B )  e.  ran  ( (
DIsoH `  K ) `  W ) )
773, 4, 5, 6, 7, 8, 9, 70, 76dochdmm1 35024 . . 3  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  ( ( N `
 { X }
)  .(+)  ( N `  { Y } ) ) ) ( (joinH `  K ) `  W
) (  ._|_  `  (  ._|_  `  B ) ) ) )
7857fveq2d 5896 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) )
793, 5, 7, 6, 10, 9, 28dochocsp 34993 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( N `
 { X ,  Y } ) )  =  (  ._|_  `  { X ,  Y } ) )
8078, 79eqtr3d 2498 . . . 4  |-  ( ph  ->  (  ._|_  `  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) ) )  =  (  ._|_  `  { X ,  Y } ) )
813, 5, 4, 71dih1dimat 34944 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  e.  A
)  ->  B  e.  ran  ( ( DIsoH `  K
) `  W )
)
829, 73, 81syl2anc 671 . . . . 5  |-  ( ph  ->  B  e.  ran  (
( DIsoH `  K ) `  W ) )
833, 4, 7dochoc 34981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  B  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  B ) )  =  B )
849, 82, 83syl2anc 671 . . . 4  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  B ) )  =  B )
8580, 84oveq12d 6338 . . 3  |-  ( ph  ->  ( (  ._|_  `  (
( N `  { X } )  .(+)  ( N `
 { Y }
) ) ) ( (joinH `  K ) `  W ) (  ._|_  `  (  ._|_  `  B ) ) )  =  ( (  ._|_  `  { X ,  Y } ) ( (joinH `  K ) `  W ) B ) )
863, 4, 5, 6, 7dochcl 34967 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { X ,  Y }  C_  V )  ->  (  ._|_  `  { X ,  Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
879, 28, 86syl2anc 671 . . . 4  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
883, 4, 8, 5, 26, 71, 9, 87, 73dihjat2 35045 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( (joinH `  K ) `  W
) B )  =  ( (  ._|_  `  { X ,  Y }
)  .(+)  B ) )
8977, 85, 883eqtrd 2500 . 2  |-  ( ph  ->  (  ._|_  `  ( ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  i^i  (  ._|_  `  B ) ) )  =  ( ( 
._|_  `  { X ,  Y } )  .(+)  B ) )
903, 5, 7, 6, 10, 9, 20dochocsp 34993 . 2  |-  ( ph  ->  (  ._|_  `  ( N `
 { ( X 
.+  Y ) } ) )  =  ( 
._|_  `  { ( X 
.+  Y ) } ) )
9169, 89, 903eqtr3d 2504 1  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
)  .(+)  B )  =  (  ._|_  `  { ( X  .+  Y ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633    \ cdif 3413    i^i cin 3415    C_ wss 3416   {csn 3980   {cpr 3982   ran crn 4857   ` cfv 5605  (class class class)co 6320   Basecbs 15176   +g cplusg 15245   0gc0g 15393  SubGrpcsubg 16866   LSSumclsm 17341   LModclmod 18146   LSubSpclss 18210   LSpanclspn 18249  LSAtomsclsa 32586   HLchlt 32962   LHypclh 33595   DVecHcdvh 34692   DIsoHcdih 34842   ocHcoch 34961  joinHcdjh 35008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-riotaBAD 32571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-undef 7051  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-sca 15261  df-vsca 15262  df-0g 15395  df-mre 15547  df-mrc 15548  df-acs 15550  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-submnd 16638  df-grp 16728  df-minusg 16729  df-sbg 16730  df-subg 16869  df-cntz 17026  df-oppg 17052  df-lsm 17343  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-dvr 17966  df-drng 18032  df-lmod 18148  df-lss 18211  df-lsp 18250  df-lvec 18381  df-lsatoms 32588  df-lshyp 32589  df-lcv 32631  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963  df-llines 33109  df-lplanes 33110  df-lvols 33111  df-lines 33112  df-psubsp 33114  df-pmap 33115  df-padd 33407  df-lhyp 33599  df-laut 33600  df-ldil 33715  df-ltrn 33716  df-trl 33771  df-tgrp 34356  df-tendo 34368  df-edring 34370  df-dveca 34616  df-disoa 34643  df-dvech 34693  df-dib 34753  df-dic 34787  df-dih 34843  df-doch 34962  df-djh 35009
This theorem is referenced by:  lcfrlem25  35181  lcfrlem35  35191
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