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Theorem cdlemn11pre 30530
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 30527, cdlemn11b 30528, cdlemn11c 30529, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn11a.b  |-  B  =  ( Base `  K
)
cdlemn11a.l  |-  .<_  =  ( le `  K )
cdlemn11a.j  |-  .\/  =  ( join `  K )
cdlemn11a.a  |-  A  =  ( Atoms `  K )
cdlemn11a.h  |-  H  =  ( LHyp `  K
)
cdlemn11a.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn11a.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
cdlemn11a.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn11a.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemn11a.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdlemn11a.i  |-  I  =  ( ( DIsoB `  K
) `  W )
cdlemn11a.J  |-  J  =  ( ( DIsoC `  K
) `  W )
cdlemn11a.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn11a.d  |-  .+  =  ( +g  `  U )
cdlemn11a.s  |-  .(+)  =  (
LSSum `  U )
cdlemn11a.f  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
cdlemn11a.g  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  N )
Assertion
Ref Expression
cdlemn11pre  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  N  .<_  ( Q  .\/  X ) )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    h, N    P, h    Q, h    T, h   
h, W
Allowed substitution hints:    .+ ( h)    .(+) ( h)    R( h)    U( h)    E( h)    F( h)    G( h)    I( h)    J( h)    .\/ ( h)    O( h)    X( h)

Proof of Theorem cdlemn11pre
StepHypRef Expression
1 cdlemn11a.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemn11a.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemn11a.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemn11a.a . . 3  |-  A  =  ( Atoms `  K )
5 cdlemn11a.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdlemn11a.p . . 3  |-  P  =  ( ( oc `  K ) `  W
)
7 cdlemn11a.o . . 3  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
8 cdlemn11a.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemn11a.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
10 cdlemn11a.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
11 cdlemn11a.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
12 cdlemn11a.J . . 3  |-  J  =  ( ( DIsoC `  K
) `  W )
13 cdlemn11a.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
14 cdlemn11a.d . . 3  |-  .+  =  ( +g  `  U )
15 cdlemn11a.s . . 3  |-  .(+)  =  (
LSSum `  U )
16 cdlemn11a.f . . 3  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
17 cdlemn11a.g . . 3  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  N )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemn11c 30529 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  E. y  e.  ( J `  Q
) E. z  e.  ( I `  X
) <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) )
19 simp1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 simp21 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
212, 4, 5, 6, 8, 10, 12, 16dicelval3 30500 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( y  e.  ( J `  Q )  <->  E. s  e.  E  y  =  <. ( s `
 F ) ,  s >. ) )
2219, 20, 21syl2anc 645 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( y  e.  ( J `  Q
)  <->  E. s  e.  E  y  =  <. ( s `
 F ) ,  s >. ) )
23 simp23 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( X  e.  B  /\  X  .<_  W ) )
241, 2, 5, 8, 9, 7, 11dibelval3 30467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
z  e.  ( I `
 X )  <->  E. g  e.  T  ( z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) ) )
2519, 23, 24syl2anc 645 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( z  e.  ( I `  X
)  <->  E. g  e.  T  ( z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) ) )
2622, 25anbi12d 694 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( (
y  e.  ( J `
 Q )  /\  z  e.  ( I `  X ) )  <->  ( E. s  e.  E  y  =  <. ( s `  F ) ,  s
>.  /\  E. g  e.  T  ( z  = 
<. g ,  O >.  /\  ( R `  g
)  .<_  X ) ) ) )
27 reeanv 2678 . . . . 5  |-  ( E. s  e.  E  E. g  e.  T  (
y  =  <. (
s `  F ) ,  s >.  /\  (
z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) )  <->  ( E. s  e.  E  y  =  <. ( s `  F ) ,  s
>.  /\  E. g  e.  T  ( z  = 
<. g ,  O >.  /\  ( R `  g
)  .<_  X ) ) )
28 simpl1 963 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
29 simpl21 1038 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
30 simpl22 1039 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( N  e.  A  /\  -.  N  .<_  W ) )
31 simpl23 1040 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( X  e.  B  /\  X  .<_  W ) )
32 simpr1r 1018 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  g  e.  T )
33 simpr1l 1017 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  s  e.  E )
34 simpr3 968 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
)
351, 2, 4, 5, 6, 7, 8, 10, 13, 14, 16, 17cdlemn9 30525 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  (
g `  Q )  =  N )
3628, 29, 30, 33, 32, 34, 35syl123anc 1204 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  (
g `  Q )  =  N )
37 simpr2 967 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( R `  g )  .<_  X )
381, 2, 3, 4, 5, 8, 9cdlemn10 30526 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  N  /\  ( R `  g ) 
.<_  X ) )  ->  N  .<_  ( Q  .\/  X ) )
3928, 29, 30, 31, 32, 36, 37, 38syl133anc 1210 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  (
( J `  Q
)  .(+)  ( I `  X ) ) )  /\  ( ( s  e.  E  /\  g  e.  T )  /\  ( R `  g )  .<_  X  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  N  .<_  ( Q  .\/  X
) )
40393exp2 1174 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( (
s  e.  E  /\  g  e.  T )  ->  ( ( R `  g )  .<_  X  -> 
( <. G ,  (  _I  |`  T ) >.  =  ( <. (
s `  F ) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) ) ) )
41 oveq12 5766 . . . . . . . . . . . . . 14  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( y  .+  z )  =  (
<. ( s `  F
) ,  s >.  .+  <. g ,  O >. ) )
4241eqeq2d 2267 . . . . . . . . . . . . 13  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( <. G ,  (  _I  |`  T )
>.  =  ( y  .+  z )  <->  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )
4342imbi1d 310 . . . . . . . . . . . 12  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( ( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q 
.\/  X ) )  <-> 
( <. G ,  (  _I  |`  T ) >.  =  ( <. (
s `  F ) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) ) )
4443imbi2d 309 . . . . . . . . . . 11  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( (
( R `  g
)  .<_  X  ->  ( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q 
.\/  X ) ) )  <->  ( ( R `
 g )  .<_  X  ->  ( <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) ) ) )
4544biimprd 216 . . . . . . . . . 10  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( (
( R `  g
)  .<_  X  ->  ( <. G ,  (  _I  |`  T ) >.  =  (
<. ( s `  F
) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) )  ->  (
( R `  g
)  .<_  X  ->  ( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q 
.\/  X ) ) ) ) )
4645com23 74 . . . . . . . . 9  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( ( R `  g )  .<_  X  ->  ( (
( R `  g
)  .<_  X  ->  ( <. G ,  (  _I  |`  T ) >.  =  (
<. ( s `  F
) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) )  ->  ( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q 
.\/  X ) ) ) ) )
4746impr 605 . . . . . . . 8  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  (
z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) )  -> 
( ( ( R `
 g )  .<_  X  ->  ( <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) )  ->  ( <. G , 
(  _I  |`  T )
>.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) ) )
4847com12 29 . . . . . . 7  |-  ( ( ( R `  g
)  .<_  X  ->  ( <. G ,  (  _I  |`  T ) >.  =  (
<. ( s `  F
) ,  s >.  .+  <. g ,  O >. )  ->  N  .<_  ( Q  .\/  X ) ) )  ->  (
( y  =  <. ( s `  F ) ,  s >.  /\  (
z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) )  -> 
( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) ) )
4940, 48syl6 31 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( (
s  e.  E  /\  g  e.  T )  ->  ( ( y  = 
<. ( s `  F
) ,  s >.  /\  ( z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) )  -> 
( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) ) ) )
5049rexlimdvv 2644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( E. s  e.  E  E. g  e.  T  (
y  =  <. (
s `  F ) ,  s >.  /\  (
z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) )  -> 
( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) ) )
5127, 50syl5bir 211 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( ( E. s  e.  E  y  =  <. ( s `
 F ) ,  s >.  /\  E. g  e.  T  ( z  =  <. g ,  O >.  /\  ( R `  g )  .<_  X ) )  ->  ( <. G ,  (  _I  |`  T )
>.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) ) )
5226, 51sylbid 208 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( (
y  e.  ( J `
 Q )  /\  z  e.  ( I `  X ) )  -> 
( <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) ) )
5352rexlimdvv 2644 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( E. y  e.  ( J `  Q ) E. z  e.  ( I `  X
) <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z )  ->  N  .<_  ( Q  .\/  X
) ) )
5418, 53mpd 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  N  .<_  ( Q  .\/  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2517    C_ wss 3094   <.cop 3584   class class class wbr 3963    e. cmpt 4017    _I cid 4241    |` cres 4628   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075   +g cplusg 13135   lecple 13142   occoc 13143   joincjn 14005   LSSumclsm 14872   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477   TEndoctendo 30071   DVecHcdvh 30398   DIsoBcdib 30458   DIsoCcdic 30492
This theorem is referenced by:  cdlemn11  30531
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-tpos 6133  df-iota 6190  df-undef 6229  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-n0 9898  df-z 9957  df-uz 10163  df-fz 10714  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-sca 13151  df-vsca 13152  df-0g 13331  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-mnd 14294  df-grp 14416  df-minusg 14417  df-sbg 14418  df-subg 14545  df-lsm 14874  df-mgp 15253  df-ring 15267  df-ur 15269  df-oppr 15332  df-dvdsr 15350  df-unit 15351  df-invr 15381  df-dvr 15392  df-drng 15441  df-lmod 15556  df-lss 15617  df-lvec 15783  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478  df-tendo 30074  df-edring 30076  df-disoa 30349  df-dvech 30399  df-dib 30459  df-dic 30493
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