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Theorem cdlemk50 31214
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120.  G,  I stand for g, h.  X represents tau. TODO: Combine into cdlemk52 31216? (Contributed by NM, 23-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
Assertion
Ref Expression
cdlemk50  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    H, b,
g, z    K, b,
g, z    N, b,
g, z    P, b,
z    R, b, z    T, b, z    W, b, g, z    z, Y    G, b    I, b, g, z
Allowed substitution hints:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk50
StepHypRef Expression
1 cdlemk5.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemk5.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemk5.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemk5.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemk5.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemk5.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemk5.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk5.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk5.z . . 3  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
10 cdlemk5.y . . 3  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
11 cdlemk5.x . . 3  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemk49 31213 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemk48 31212 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )
14 simp11l 1066 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  K  e.  HL )
15 hllat 29626 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1614, 15syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  K  e.  Lat )
17 simp11 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 simp12 986 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
19 simp13 987 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )
20 simp21 988 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  N  e.  T )
21 simp22 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
22 simp23 990 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( R `  F )  =  ( R `  N ) )
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemk35s 31199 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ G  /  g ]_ X  e.  T )
2417, 18, 19, 20, 21, 22, 23syl132anc 1200 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  [_ G  /  g ]_ X  e.  T )
25 simp3 957 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemk35s 31199 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ I  /  g ]_ X  e.  T )
2717, 18, 25, 20, 21, 22, 26syl132anc 1200 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  [_ I  /  g ]_ X  e.  T )
286, 7ltrnco 30981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T  /\  [_ I  /  g ]_ X  e.  T
)  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  e.  T
)
2917, 24, 27, 28syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
)  e.  T )
30 simp22l 1074 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  P  e.  A )
312, 5, 6, 7ltrnat 30402 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  e.  T  /\  P  e.  A
)  ->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A )
3217, 29, 30, 31syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A )
331, 5atbase 29552 . . . 4  |-  ( ( ( [_ G  / 
g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A  ->  ( (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  B )
3432, 33syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  B )
352, 5, 6, 7ltrnat 30402 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T  /\  P  e.  A
)  ->  ( [_ G  /  g ]_ X `  P )  e.  A
)
3617, 24, 30, 35syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ G  /  g ]_ X `  P )  e.  A )
371, 5atbase 29552 . . . . 5  |-  ( (
[_ G  /  g ]_ X `  P )  e.  A  ->  ( [_ G  /  g ]_ X `  P )  e.  B )
3836, 37syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ G  /  g ]_ X `  P )  e.  B )
391, 6, 7, 8trlcl 30426 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ I  / 
g ]_ X  e.  T
)  ->  ( R `  [_ I  /  g ]_ X )  e.  B
)
4017, 27, 39syl2anc 642 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( R `  [_ I  / 
g ]_ X )  e.  B )
411, 3latjcl 14158 . . . 4  |-  ( ( K  e.  Lat  /\  ( [_ G  /  g ]_ X `  P )  e.  B  /\  ( R `  [_ I  / 
g ]_ X )  e.  B )  ->  (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  [_ I  /  g ]_ X ) )  e.  B )
4216, 38, 40, 41syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  [_ I  /  g ]_ X ) )  e.  B )
432, 5, 6, 7ltrnat 30402 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ I  / 
g ]_ X  e.  T  /\  P  e.  A
)  ->  ( [_ I  /  g ]_ X `  P )  e.  A
)
4417, 27, 30, 43syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ I  /  g ]_ X `  P )  e.  A )
451, 5atbase 29552 . . . . 5  |-  ( (
[_ I  /  g ]_ X `  P )  e.  A  ->  ( [_ I  /  g ]_ X `  P )  e.  B )
4644, 45syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ I  /  g ]_ X `  P )  e.  B )
471, 6, 7, 8trlcl 30426 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T
)  ->  ( R `  [_ G  /  g ]_ X )  e.  B
)
4817, 24, 47syl2anc 642 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  e.  B )
491, 3latjcl 14158 . . . 4  |-  ( ( K  e.  Lat  /\  ( [_ I  /  g ]_ X `  P )  e.  B  /\  ( R `  [_ G  / 
g ]_ X )  e.  B )  ->  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  [_ G  /  g ]_ X ) )  e.  B )
5016, 46, 48, 49syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  [_ G  /  g ]_ X ) )  e.  B )
511, 2, 4latlem12 14186 . . 3  |-  ( ( K  e.  Lat  /\  ( ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  e.  B  /\  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) )  e.  B  /\  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  [_ G  /  g ]_ X ) )  e.  B ) )  -> 
( ( ( (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) )  /\  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )  <->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( ( (
[_ G  /  g ]_ X `  P ) 
.\/  ( R `  [_ I  /  g ]_ X ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) ) ) )
5216, 34, 42, 50, 51syl13anc 1184 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) )  /\  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )  <->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( ( (
[_ G  /  g ]_ X `  P ) 
.\/  ( R `  [_ I  /  g ]_ X ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) ) ) )
5312, 13, 52mpbi2and 887 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   [_csb 3083   class class class wbr 4025    _I cid 4306   `'ccnv 4690    |` cres 4693    o. ccom 4695   ` cfv 5257  (class class class)co 5860   iota_crio 6299   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   Latclat 14153   Atomscatm 29526   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   trLctrl 30420
This theorem is referenced by:  cdlemk52  31216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421
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