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Theorem dihjat1lem 37298
Description: Subspace sum of a closed subspace and an atom. (pmapjat1 35720 analog.) TODO: merge into dihjat1 37299? (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 6311 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
31oveq1d 6311 . . . 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 3622 . . . . . . . . 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 37201 . . . . . . . 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 37283 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( N `  { T } ) )
184, 5, 9dvhlmod 36980 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
19 eqid 2457 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
2013, 19, 14lspsncl 17750 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  T  e.  V )  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2118, 12, 20syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2219lsssubg 17730 . . . . . . . 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 16817 . . . . . . 7  |-  ( ( N `  { T } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { T }
) )  =  ( N `  { T } ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  { T } ) )  =  ( N `  { T } ) )
2717, 26eqtr4d 2501 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
2827adantr 465 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
293, 28eqtr4d 2501 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
302, 29eqtr4d 2501 . 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 37148 . . . . . . . 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 17710 . . . . . . . 8  |-  ( ( N `  { T } )  e.  (
LSubSp `  U )  -> 
( N `  { T } )  C_  V
)
3621, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  C_  V
)
374, 7, 5, 13, 8djhcl 37270 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  V  /\  ( N `  { T } )  C_  V ) )  -> 
( X  .\/  ( N `  { T } ) )  e. 
ran  I )
389, 34, 36, 37syl12anc 1226 . . . . . 6  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )
394, 5, 7, 13dihrnss 37148 . . . . . 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 37147 . . . . . . 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 17856 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  ( LSubSp `  U )  /\  ( N `  { T } )  e.  (
LSubSp `  U ) )  ->  ( X  .(+)  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
4518, 43, 21, 44syl3anc 1228 . . . . 5  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  e.  (
LSubSp `  U ) )
4645adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  e.  (
LSubSp `  U ) )
47 simplr 755 . . . . . . . 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 37285 . . . . . . . 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 765 . . . . . . . . . . 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 3622 . . . . . . . . . . . . 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 17768 . . . . . . . . . . 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 17766 . . . . . . . . . . . 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 766 . . . . . . . . . . 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 4042 . . . . . . . . . . . . . . 15  |-  ( z  =  T  ->  { z }  =  { T } )
6665fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( z  =  T  ->  ( N `  { z } )  =  ( N `  { T } ) )
6766oveq2d 6312 . . . . . . . . . . . . 13  |-  ( z  =  T  ->  (
( N `  {
y } )  .\/  ( N `  { z } ) )  =  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) )
6867sseq2d 3527 . . . . . . . . . . . 12  |-  ( z  =  T  ->  (
( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { T }
) ) ) )
6968rspcev 3210 . . . . . . . . . . 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 2928 . . . . . . 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 37147 . . . . . . . 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 17767 . . . . . 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 17858 . . . . . . 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 755 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
y  e.  X )
8413, 19lssel 17711 . . . . . . . . . . . . 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 17711 . . . . . . . . . . . . 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 37297 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { y } ) 
.(+)  ( N `  { z } ) )  =  ( ( N `  { y } )  .\/  ( N `  { z } ) ) )
9190sseq2d 3527 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { x } ) 
C_  ( ( N `
 { y } )  .(+)  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { z } ) ) ) )
9291rexbidva 2965 . . . . . . . . 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 2965 . . . . . . . 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 17726 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  C_  ( X  .(+)  ( N `
 { T }
) ) )
994, 5, 13, 24, 8, 9, 34, 36djhsumss 37277 . . . 4  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10099adantr 465 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10198, 100eqssd 3516 . 2  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
10230, 101pm2.61dane 2775 1  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    \ cdif 3468    C_ wss 3471   {csn 4032   ran crn 5009   ` cfv 5594  (class class class)co 6296   Basecbs 14644   0gc0g 14857  SubGrpcsubg 16322   LSSumclsm 16781   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744   HLchlt 35218   LHypclh 35851   DVecHcdvh 36948   DIsoHcdih 37098  joinHcdjh 37264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-undef 7020  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-0g 14859  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-cntz 16482  df-lsm 16783  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-drng 17525  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lvec 17876  df-lsatoms 34844  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027  df-tgrp 36612  df-tendo 36624  df-edring 36626  df-dveca 36872  df-disoa 36899  df-dvech 36949  df-dib 37009  df-dic 37043  df-dih 37099  df-doch 37218  df-djh 37265
This theorem is referenced by:  dihjat1  37299
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