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Theorem dihjat1lem 34966
Description: Subspace sum of a closed subspace and an atom. (pmapjat1 33390 analog.) TODO: merge into dihjat1 34967? (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 6101 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
31oveq1d 6101 . . . 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 3473 . . . . . . . . 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 34869 . . . . . . . 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 34951 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( N `  { T } ) )
184, 5, 9dvhlmod 34648 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
19 eqid 2438 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
2013, 19, 14lspsncl 17038 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  T  e.  V )  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2118, 12, 20syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2219lsssubg 17018 . . . . . . . 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 16160 . . . . . . 7  |-  ( ( N `  { T } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { T }
) )  =  ( N `  { T } ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  { T } ) )  =  ( N `  { T } ) )
2717, 26eqtr4d 2473 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
2827adantr 465 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
293, 28eqtr4d 2473 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
302, 29eqtr4d 2473 . 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 34816 . . . . . . . 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 16998 . . . . . . . 8  |-  ( ( N `  { T } )  e.  (
LSubSp `  U )  -> 
( N `  { T } )  C_  V
)
3621, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  C_  V
)
374, 7, 5, 13, 8djhcl 34938 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  V  /\  ( N `  { T } )  C_  V ) )  -> 
( X  .\/  ( N `  { T } ) )  e. 
ran  I )
389, 34, 36, 37syl12anc 1216 . . . . . 6  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )
394, 5, 7, 13dihrnss 34816 . . . . . 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 34815 . . . . . . 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 17144 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  ( LSubSp `  U )  /\  ( N `  { T } )  e.  (
LSubSp `  U ) )  ->  ( X  .(+)  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
4518, 43, 21, 44syl3anc 1218 . . . . 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 34953 . . . . . . . 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 3473 . . . . . . . . . . . . 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 17056 . . . . . . . . . . 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 17054 . . . . . . . . . . . 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 3882 . . . . . . . . . . . . . . 15  |-  ( z  =  T  ->  { z }  =  { T } )
6665fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( z  =  T  ->  ( N `  { z } )  =  ( N `  { T } ) )
6766oveq2d 6102 . . . . . . . . . . . . 13  |-  ( z  =  T  ->  (
( N `  {
y } )  .\/  ( N `  { z } ) )  =  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) )
6867sseq2d 3379 . . . . . . . . . . . 12  |-  ( z  =  T  ->  (
( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { T }
) ) ) )
6968rspcev 3068 . . . . . . . . . . 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 2820 . . . . . . 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 34815 . . . . . . . 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 17055 . . . . . 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 17146 . . . . . . 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 16999 . . . . . . . . . . . . 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 16999 . . . . . . . . . . . . 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 34965 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { y } ) 
.(+)  ( N `  { z } ) )  =  ( ( N `  { y } )  .\/  ( N `  { z } ) ) )
9190sseq2d 3379 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { x } ) 
C_  ( ( N `
 { y } )  .(+)  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { z } ) ) ) )
9291rexbidva 2727 . . . . . . . . 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 2727 . . . . . . . 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 17014 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  C_  ( X  .(+)  ( N `
 { T }
) ) )
994, 5, 13, 24, 8, 9, 34, 36djhsumss 34945 . . . 4  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10099adantr 465 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10198, 100eqssd 3368 . 2  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
10230, 101pm2.61dane 2684 1  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711    \ cdif 3320    C_ wss 3323   {csn 3872   ran crn 4836   ` cfv 5413  (class class class)co 6086   Basecbs 14166   0gc0g 14370  SubGrpcsubg 15666   LSSumclsm 16124   LModclmod 16928   LSubSpclss 16993   LSpanclspn 17032   HLchlt 32888   LHypclh 33521   DVecHcdvh 34616   DIsoHcdih 34766  joinHcdjh 34932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-riotaBAD 32497
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-undef 6784  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-0g 14372  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-lmod 16930  df-lss 16994  df-lsp 17033  df-lvec 17164  df-lsatoms 32514  df-oposet 32714  df-ol 32716  df-oml 32717  df-covers 32804  df-ats 32805  df-atl 32836  df-cvlat 32860  df-hlat 32889  df-llines 33035  df-lplanes 33036  df-lvols 33037  df-lines 33038  df-psubsp 33040  df-pmap 33041  df-padd 33333  df-lhyp 33525  df-laut 33526  df-ldil 33641  df-ltrn 33642  df-trl 33696  df-tgrp 34280  df-tendo 34292  df-edring 34294  df-dveca 34540  df-disoa 34567  df-dvech 34617  df-dib 34677  df-dic 34711  df-dih 34767  df-doch 34886  df-djh 34933
This theorem is referenced by:  dihjat1  34967
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