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Theorem cdlemn11pre 34543
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 34540, cdlemn11b 34541, cdlemn11c 34542, 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
Dummy variables  g 
s  y  z are mutually distinct and distinct from all other variables.
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 34542 . 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 983 . . . . . 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 1016 . . . . . 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 34513 . . . . . 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 656 . . . . 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 1018 . . . . . 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 34480 . . . . . 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 656 . . . . 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 705 . . . 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 2886 . . . . 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 986 . . . . . . . . 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 1061 . . . . . . . . 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 1062 . . . . . . . . 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 1063 . . . . . . . . 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 1041 . . . . . . . . 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 1040 . . . . . . . . . 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 991 . . . . . . . . . 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 34538 . . . . . . . . . 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 1230 . . . . . . . . 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 990 . . . . . . . . 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 34539 . . . . . . . . 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 1236 . . . . . . . 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 1200 . . . . . . 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 6099 . . . . . . . . . . . . . 14  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( y  .+  z )  =  (
<. ( s `  F
) ,  s >.  .+  <. g ,  O >. ) )
4241eqeq2d 2452 . . . . . . . . . . . . 13  |-  ( ( y  =  <. (
s `  F ) ,  s >.  /\  z  =  <. g ,  O >. )  ->  ( <. G ,  (  _I  |`  T )
>.  =  ( y  .+  z )  <->  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )
4342imbi1d 317 . . . . . . . . . . . 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 316 . . . . . . . . . . 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 223 . . . . . . . . . 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 78 . . . . . . . . 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 616 . . . . . . . 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 31 . . . . . . 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 33 . . . . . 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 2845 . . . . 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 218 . . . 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 215 . . 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 2845 . 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 15 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347    _I cid 4627    |` cres 4838   ` cfv 5415   iota_crio 6048  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   lecple 14241   occoc 14242   joincjn 15110   LSSumclsm 16126   Atomscatm 32596   HLchlt 32683   LHypclh 33316   LTrncltrn 33433   trLctrl 33490   TEndoctendo 34084   DVecHcdvh 34411   DIsoBcdib 34471   DIsoCcdic 34505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32292
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-0g 14376  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-lsm 16128  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 16992  df-lvec 17162  df-oposet 32509  df-ol 32511  df-oml 32512  df-covers 32599  df-ats 32600  df-atl 32631  df-cvlat 32655  df-hlat 32684  df-llines 32830  df-lplanes 32831  df-lvols 32832  df-lines 32833  df-psubsp 32835  df-pmap 32836  df-padd 33128  df-lhyp 33320  df-laut 33321  df-ldil 33436  df-ltrn 33437  df-trl 33491  df-tendo 34087  df-edring 34089  df-disoa 34362  df-dvech 34412  df-dib 34472  df-dic 34506
This theorem is referenced by:  cdlemn11  34544
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