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Theorem dihjat1lem 35396
Description: Subspace sum of a closed subspace and an atom. (pmapjat1 33820 analog.) TODO: merge into dihjat1 35397? (Contributed by NM, 18-Aug-2014.)
Hypotheses
Ref Expression
dihjat1.h  |-  H  =  ( LHyp `  K
)
dihjat1.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjat1.v  |-  V  =  ( Base `  U
)
dihjat1.p  |-  .(+)  =  (
LSSum `  U )
dihjat1.n  |-  N  =  ( LSpan `  U )
dihjat1.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihjat1.j  |-  .\/  =  ( (joinH `  K ) `  W )
dihjat1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dihjat1.x  |-  ( ph  ->  X  e.  ran  I
)
dihjat1.o  |-  .0.  =  ( 0g `  U )
dihjat1lem.q  |-  ( ph  ->  T  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
dihjat1lem  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )

Proof of Theorem dihjat1lem
Dummy variables  y  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  ->  X  =  {  .0.  } )
21oveq1d 6214 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
31oveq1d 6214 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
4 dihjat1.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
5 dihjat1.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
6 dihjat1.o . . . . . . 7  |-  .0.  =  ( 0g `  U )
7 dihjat1.i . . . . . . 7  |-  I  =  ( ( DIsoH `  K
) `  W )
8 dihjat1.j . . . . . . 7  |-  .\/  =  ( (joinH `  K ) `  W )
9 dihjat1.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 dihjat1lem.q . . . . . . . . 9  |-  ( ph  ->  T  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3585 . . . . . . . . 9  |-  ( T  e.  ( V  \  {  .0.  } )  ->  T  e.  V )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  T  e.  V )
13 dihjat1.v . . . . . . . . 9  |-  V  =  ( Base `  U
)
14 dihjat1.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
154, 5, 13, 14, 7dihlsprn 35299 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  T  e.  V
)  ->  ( N `  { T } )  e.  ran  I )
169, 12, 15syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  e.  ran  I )
174, 5, 6, 7, 8, 9, 16djh02 35381 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( N `  { T } ) )
184, 5, 9dvhlmod 35078 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
19 eqid 2454 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
2013, 19, 14lspsncl 17180 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  T  e.  V )  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2118, 12, 20syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2219lsssubg 17160 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  ( N `  { T } )  e.  (
LSubSp `  U ) )  ->  ( N `  { T } )  e.  (SubGrp `  U )
)
2318, 21, 22syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  e.  (SubGrp `  U ) )
24 dihjat1.p . . . . . . . 8  |-  .(+)  =  (
LSSum `  U )
256, 24lsm02 16289 . . . . . . 7  |-  ( ( N `  { T } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { T }
) )  =  ( N `  { T } ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  { T } ) )  =  ( N `  { T } ) )
2717, 26eqtr4d 2498 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
2827adantr 465 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
293, 28eqtr4d 2498 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
302, 29eqtr4d 2498 . 2  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
3118adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  ->  U  e.  LMod )
32 dihjat1.x . . . . . . . 8  |-  ( ph  ->  X  e.  ran  I
)
334, 5, 7, 13dihrnss 35246 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  C_  V )
349, 32, 33syl2anc 661 . . . . . . 7  |-  ( ph  ->  X  C_  V )
3513, 19lssss 17140 . . . . . . . 8  |-  ( ( N `  { T } )  e.  (
LSubSp `  U )  -> 
( N `  { T } )  C_  V
)
3621, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  C_  V
)
374, 7, 5, 13, 8djhcl 35368 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  V  /\  ( N `  { T } )  C_  V ) )  -> 
( X  .\/  ( N `  { T } ) )  e. 
ran  I )
389, 34, 36, 37syl12anc 1217 . . . . . 6  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )
394, 5, 7, 13dihrnss 35246 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )  -> 
( X  .\/  ( N `  { T } ) )  C_  V )
409, 38, 39syl2anc 661 . . . . 5  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  C_  V )
4140adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  C_  V )
424, 5, 7, 19dihrnlss 35245 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  e.  ( LSubSp `  U )
)
439, 32, 42syl2anc 661 . . . . . 6  |-  ( ph  ->  X  e.  ( LSubSp `  U ) )
4419, 24lsmcl 17286 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  ( LSubSp `  U )  /\  ( N `  { T } )  e.  (
LSubSp `  U ) )  ->  ( X  .(+)  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
4518, 43, 21, 44syl3anc 1219 . . . . 5  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  e.  (
LSubSp `  U ) )
4645adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  e.  (
LSubSp `  U ) )
47 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  X  =/=  {  .0.  } )
489ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4932ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  X  e.  ran  I )
50 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  x  e.  ( V  \  {  .0.  } ) )
5110ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  T  e.  ( V  \  {  .0.  } ) )
524, 5, 13, 6, 14, 7, 8, 48, 49, 50, 51djhcvat42 35383 . . . . . . . 8  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
( X  =/=  {  .0.  }  /\  ( N `
 { x }
)  C_  ( X  .\/  ( N `  { T } ) ) )  ->  E. y  e.  ( V  \  {  .0.  } ) ( ( N `
 { y } )  C_  X  /\  ( N `  { x } )  C_  (
( N `  {
y } )  .\/  ( N `  { T } ) ) ) ) )
5347, 52mpand 675 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
( N `  {
x } )  C_  ( X  .\/  ( N `
 { T }
) )  ->  E. y  e.  ( V  \  {  .0.  } ) ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )
54 simprrl 763 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  -> 
( N `  {
y } )  C_  X )
5518ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  ->  U  e.  LMod )
5643ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  ->  X  e.  ( LSubSp `  U ) )
57 eldifi 3585 . . . . . . . . . . . . 13  |-  ( y  e.  ( V  \  {  .0.  } )  -> 
y  e.  V )
5857ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  -> 
y  e.  V )
5913, 19, 14, 55, 56, 58lspsnel5 17198 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  -> 
( y  e.  X  <->  ( N `  { y } )  C_  X
) )
6054, 59mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  -> 
y  e.  X )
6112ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  ->  T  e.  V )
6213, 14lspsnid 17196 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  T  e.  V )  ->  T  e.  ( N `  { T } ) )
6355, 61, 62syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  ->  T  e.  ( N `  { T } ) )
64 simprrr 764 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  -> 
( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) )
65 sneq 3994 . . . . . . . . . . . . . . 15  |-  ( z  =  T  ->  { z }  =  { T } )
6665fveq2d 5802 . . . . . . . . . . . . . 14  |-  ( z  =  T  ->  ( N `  { z } )  =  ( N `  { T } ) )
6766oveq2d 6215 . . . . . . . . . . . . 13  |-  ( z  =  T  ->  (
( N `  {
y } )  .\/  ( N `  { z } ) )  =  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) )
6867sseq2d 3491 . . . . . . . . . . . 12  |-  ( z  =  T  ->  (
( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { T }
) ) ) )
6968rspcev 3177 . . . . . . . . . . 11  |-  ( ( T  e.  ( N `
 { T }
)  /\  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { T }
) ) )  ->  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) )
7063, 64, 69syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  ->  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) )
7160, 70jca 532 . . . . . . . . 9  |-  ( ( ( ( ph  /\  X  =/=  {  .0.  }
)  /\  x  e.  ( V  \  {  .0.  } ) )  /\  (
y  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) ) )  -> 
( y  e.  X  /\  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) )
7271ex 434 . . . . . . . 8  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
( y  e.  ( V  \  {  .0.  } )  /\  ( ( N `  { y } )  C_  X  /\  ( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) ) )  ->  (
y  e.  X  /\  E. z  e.  ( N `
 { T }
) ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { z } ) ) ) ) )
7372reximdv2 2929 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( E. y  e.  ( V  \  {  .0.  }
) ( ( N `
 { y } )  C_  X  /\  ( N `  { x } )  C_  (
( N `  {
y } )  .\/  ( N `  { T } ) ) )  ->  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) )
7453, 73syld 44 . . . . . 6  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
( N `  {
x } )  C_  ( X  .\/  ( N `
 { T }
) )  ->  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) )
7574anim2d 565 . . . . 5  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
( x  e.  V  /\  ( N `  {
x } )  C_  ( X  .\/  ( N `
 { T }
) ) )  -> 
( x  e.  V  /\  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) ) )
764, 5, 7, 19dihrnlss 35245 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )  -> 
( X  .\/  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
779, 38, 76syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
7813, 19, 14, 18, 77lspsnel6 17197 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  .\/  ( N `
 { T }
) )  <->  ( x  e.  V  /\  ( N `  { x } )  C_  ( X  .\/  ( N `  { T } ) ) ) ) )
7978ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
x  e.  ( X 
.\/  ( N `  { T } ) )  <-> 
( x  e.  V  /\  ( N `  {
x } )  C_  ( X  .\/  ( N `
 { T }
) ) ) ) )
8013, 19, 24, 14, 18, 43, 21lsmelval2 17288 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( X  .(+)  ( N `  { T } ) )  <->  ( x  e.  V  /\  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .(+)  ( N `
 { z } ) ) ) ) )
819ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
8243ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  ->  X  e.  ( LSubSp `  U ) )
83 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
y  e.  X )
8413, 19lssel 17141 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( LSubSp `  U )  /\  y  e.  X )  ->  y  e.  V )
8582, 83, 84syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
y  e.  V )
8621ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( N `  { T } )  e.  (
LSubSp `  U ) )
87 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
z  e.  ( N `
 { T }
) )
8813, 19lssel 17141 . . . . . . . . . . . . 13  |-  ( ( ( N `  { T } )  e.  (
LSubSp `  U )  /\  z  e.  ( N `  { T } ) )  ->  z  e.  V )
8986, 87, 88syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
z  e.  V )
904, 5, 13, 24, 14, 7, 8, 81, 85, 89djhlsmat 35395 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { y } ) 
.(+)  ( N `  { z } ) )  =  ( ( N `  { y } )  .\/  ( N `  { z } ) ) )
9190sseq2d 3491 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { x } ) 
C_  ( ( N `
 { y } )  .(+)  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { z } ) ) ) )
9291rexbidva 2861 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  X )  ->  ( E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .(+)  ( N `
 { z } ) )  <->  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) )
9392rexbidva 2861 . . . . . . . 8  |-  ( ph  ->  ( E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .(+)  ( N `
 { z } ) )  <->  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) )
9493anbi2d 703 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  V  /\  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .(+)  ( N `
 { z } ) ) )  <->  ( x  e.  V  /\  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) ) )
9580, 94bitrd 253 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  .(+)  ( N `  { T } ) )  <->  ( x  e.  V  /\  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) ) )
9695ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
x  e.  ( X 
.(+)  ( N `  { T } ) )  <-> 
( x  e.  V  /\  E. y  e.  X  E. z  e.  ( N `  { T } ) ( N `
 { x }
)  C_  ( ( N `  { y } )  .\/  ( N `  { z } ) ) ) ) )
9775, 79, 963imtr4d 268 . . . 4  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  (
x  e.  ( X 
.\/  ( N `  { T } ) )  ->  x  e.  ( X  .(+)  ( N `  { T } ) ) ) )
9813, 6, 19, 31, 41, 46, 97lssssr 17156 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  C_  ( X  .(+)  ( N `
 { T }
) ) )
994, 5, 13, 24, 8, 9, 34, 36djhsumss 35375 . . . 4  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10099adantr 465 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10198, 100eqssd 3480 . 2  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
10230, 101pm2.61dane 2769 1  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799    \ cdif 3432    C_ wss 3435   {csn 3984   ran crn 4948   ` cfv 5525  (class class class)co 6199   Basecbs 14291   0gc0g 14496  SubGrpcsubg 15793   LSSumclsm 16253   LModclmod 17070   LSubSpclss 17135   LSpanclspn 17174   HLchlt 33318   LHypclh 33951   DVecHcdvh 35046   DIsoHcdih 35196  joinHcdjh 35362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-riotaBAD 32927
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-tpos 6854  df-undef 6901  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-sca 14372  df-vsca 14373  df-0g 14498  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-mnd 15533  df-submnd 15583  df-grp 15663  df-minusg 15664  df-sbg 15665  df-subg 15796  df-cntz 15953  df-lsm 16255  df-cmn 16399  df-abl 16400  df-mgp 16713  df-ur 16725  df-rng 16769  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-invr 16886  df-dvr 16897  df-drng 16956  df-lmod 17072  df-lss 17136  df-lsp 17175  df-lvec 17306  df-lsatoms 32944  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126  df-tgrp 34710  df-tendo 34722  df-edring 34724  df-dveca 34970  df-disoa 34997  df-dvech 35047  df-dib 35107  df-dic 35141  df-dih 35197  df-doch 35316  df-djh 35363
This theorem is referenced by:  dihjat1  35397
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