Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihjat1lem Structured version   Unicode version

Theorem dihjat1lem 32227
Description: Subspace sum of a closed subspace and an atom. (pmapjat1 30651 analog.) TODO: merge into dihjat1 32228? (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 449 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  ->  X  =  {  .0.  } )
21oveq1d 6097 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
31oveq1d 6097 . . . 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 3470 . . . . . . . . 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 32130 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  T  e.  V
)  ->  ( N `  { T } )  e.  ran  I )
169, 12, 15syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  e.  ran  I )
174, 5, 6, 7, 8, 9, 16djh02 32212 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( N `  { T } ) )
184, 5, 9dvhlmod 31909 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
19 eqid 2437 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
2013, 19, 14lspsncl 16054 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  T  e.  V )  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2118, 12, 20syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  e.  (
LSubSp `  U ) )
2219lsssubg 16034 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  ( N `  { T } )  e.  (
LSubSp `  U ) )  ->  ( N `  { T } )  e.  (SubGrp `  U )
)
2318, 21, 22syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  e.  (SubGrp `  U ) )
24 dihjat1.p . . . . . . . 8  |-  .(+)  =  (
LSSum `  U )
256, 24lsm02 15305 . . . . . . 7  |-  ( ( N `  { T } )  e.  (SubGrp `  U )  ->  ( {  .0.  }  .(+)  ( N `
 { T }
) )  =  ( N `  { T } ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  ->  ( {  .0.  }  .(+)  ( N `  { T } ) )  =  ( N `  { T } ) )
2717, 26eqtr4d 2472 . . . . 5  |-  ( ph  ->  ( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
2827adantr 453 . . . 4  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( {  .0.  }  .\/  ( N `  { T } ) )  =  ( {  .0.  }  .(+)  ( N `  { T } ) ) )
293, 28eqtr4d 2472 . . 3  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  =  ( {  .0.  }  .\/  ( N `  { T } ) ) )
302, 29eqtr4d 2472 . 2  |-  ( (
ph  /\  X  =  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
3118adantr 453 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  ->  U  e.  LMod )
32 dihjat1.x . . . . . . . 8  |-  ( ph  ->  X  e.  ran  I
)
334, 5, 7, 13dihrnss 32077 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  C_  V )
349, 32, 33syl2anc 644 . . . . . . 7  |-  ( ph  ->  X  C_  V )
3513, 19lssss 16014 . . . . . . . 8  |-  ( ( N `  { T } )  e.  (
LSubSp `  U )  -> 
( N `  { T } )  C_  V
)
3621, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  C_  V
)
374, 7, 5, 13, 8djhcl 32199 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  V  /\  ( N `  { T } )  C_  V ) )  -> 
( X  .\/  ( N `  { T } ) )  e. 
ran  I )
389, 34, 36, 37syl12anc 1183 . . . . . 6  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )
394, 5, 7, 13dihrnss 32077 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )  -> 
( X  .\/  ( N `  { T } ) )  C_  V )
409, 38, 39syl2anc 644 . . . . 5  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  C_  V )
4140adantr 453 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  C_  V )
424, 5, 7, 19dihrnlss 32076 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  e.  ( LSubSp `  U )
)
439, 32, 42syl2anc 644 . . . . . 6  |-  ( ph  ->  X  e.  ( LSubSp `  U ) )
4419, 24lsmcl 16156 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  ( LSubSp `  U )  /\  ( N `  { T } )  e.  (
LSubSp `  U ) )  ->  ( X  .(+)  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
4518, 43, 21, 44syl3anc 1185 . . . . 5  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  e.  (
LSubSp `  U ) )
4645adantr 453 . . . 4  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  e.  (
LSubSp `  U ) )
47 simplr 733 . . . . . . . 8  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  X  =/=  {  .0.  } )
489ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4932ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  X  e.  ran  I )
50 simpr 449 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  x  e.  ( V  \  {  .0.  } ) )
5110ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  X  =/=  {  .0.  } )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  T  e.  ( V  \  {  .0.  } ) )
524, 5, 13, 6, 14, 7, 8, 48, 49, 50, 51djhcvat42 32214 . . . . . . . 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 658 . . . . . . 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 742 . . . . . . . . . . 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 712 . . . . . . . . . . . 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 712 . . . . . . . . . . . 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 3470 . . . . . . . . . . . . 13  |-  ( y  e.  ( V  \  {  .0.  } )  -> 
y  e.  V )
5857ad2antrl 710 . . . . . . . . . . . 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 16072 . . . . . . . . . . 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 225 . . . . . . . . . 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 712 . . . . . . . . . . . 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 16070 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  T  e.  V )  ->  T  e.  ( N `  { T } ) )
6355, 61, 62syl2anc 644 . . . . . . . . . . 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 743 . . . . . . . . . . 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 3826 . . . . . . . . . . . . . . 15  |-  ( z  =  T  ->  { z }  =  { T } )
6665fveq2d 5733 . . . . . . . . . . . . . 14  |-  ( z  =  T  ->  ( N `  { z } )  =  ( N `  { T } ) )
6766oveq2d 6098 . . . . . . . . . . . . 13  |-  ( z  =  T  ->  (
( N `  {
y } )  .\/  ( N `  { z } ) )  =  ( ( N `  { y } ) 
.\/  ( N `  { T } ) ) )
6867sseq2d 3377 . . . . . . . . . . . 12  |-  ( z  =  T  ->  (
( N `  {
x } )  C_  ( ( N `  { y } ) 
.\/  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { T }
) ) ) )
6968rspcev 3053 . . . . . . . . . . 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 644 . . . . . . . . . 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 520 . . . . . . . . 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 425 . . . . . . . 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 2816 . . . . . . 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 43 . . . . . 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 550 . . . . 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 32076 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  .\/  ( N `  { T } ) )  e. 
ran  I )  -> 
( X  .\/  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
779, 38, 76syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  e.  ( LSubSp `  U )
)
7813, 19, 14, 18, 77lspsnel6 16071 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  .\/  ( N `
 { T }
) )  <->  ( x  e.  V  /\  ( N `  { x } )  C_  ( X  .\/  ( N `  { T } ) ) ) ) )
7978ad2antrr 708 . . . . 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 16158 . . . . . . 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 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
8243ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  ->  X  e.  ( LSubSp `  U ) )
83 simplr 733 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
y  e.  X )
8413, 19lssel 16015 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( LSubSp `  U )  /\  y  e.  X )  ->  y  e.  V )
8582, 83, 84syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
y  e.  V )
8621ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( N `  { T } )  e.  (
LSubSp `  U ) )
87 simpr 449 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
z  e.  ( N `
 { T }
) )
8813, 19lssel 16015 . . . . . . . . . . . . 13  |-  ( ( ( N `  { T } )  e.  (
LSubSp `  U )  /\  z  e.  ( N `  { T } ) )  ->  z  e.  V )
8986, 87, 88syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
z  e.  V )
904, 5, 13, 24, 14, 7, 8, 81, 85, 89djhlsmat 32226 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { y } ) 
.(+)  ( N `  { z } ) )  =  ( ( N `  { y } )  .\/  ( N `  { z } ) ) )
9190sseq2d 3377 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  X )  /\  z  e.  ( N `  { T } ) )  -> 
( ( N `  { x } ) 
C_  ( ( N `
 { y } )  .(+)  ( N `  { z } ) )  <->  ( N `  { x } ) 
C_  ( ( N `
 { y } )  .\/  ( N `
 { z } ) ) ) )
9291rexbidva 2723 . . . . . . . . 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 2723 . . . . . . . 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 686 . . . . . . 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 246 . . . . . 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 708 . . . . 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 261 . . . 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 16030 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  C_  ( X  .(+)  ( N `
 { T }
) ) )
994, 5, 13, 24, 8, 9, 34, 36djhsumss 32206 . . . 4  |-  ( ph  ->  ( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10099adantr 453 . . 3  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .(+)  ( N `
 { T }
) )  C_  ( X  .\/  ( N `  { T } ) ) )
10198, 100eqssd 3366 . 2  |-  ( (
ph  /\  X  =/=  {  .0.  } )  -> 
( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
10230, 101pm2.61dane 2683 1  |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `
 { T }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707    \ cdif 3318    C_ wss 3321   {csn 3815   ran crn 4880   ` cfv 5455  (class class class)co 6082   Basecbs 13470   0gc0g 13724  SubGrpcsubg 14939   LSSumclsm 15269   LModclmod 15951   LSubSpclss 16009   LSpanclspn 16048   HLchlt 30149   LHypclh 30782   DVecHcdvh 31877   DIsoHcdih 32027  joinHcdjh 32193
This theorem is referenced by:  dihjat1  32228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-undef 6544  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-0g 13728  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-mnd 14691  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-subg 14942  df-cntz 15117  df-lsm 15271  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-dvr 15789  df-drng 15838  df-lmod 15953  df-lss 16010  df-lsp 16049  df-lvec 16176  df-lsatoms 29775  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lvols 30298  df-lines 30299  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-lhyp 30786  df-laut 30787  df-ldil 30902  df-ltrn 30903  df-trl 30957  df-tgrp 31541  df-tendo 31553  df-edring 31555  df-dveca 31801  df-disoa 31828  df-dvech 31878  df-dib 31938  df-dic 31972  df-dih 32028  df-doch 32147  df-djh 32194
  Copyright terms: Public domain W3C validator