Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk50 Unicode version

Theorem cdlemk50 30391
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 30393? (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 30390 . 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 30389 . 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 1071 . . . 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 28803 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1614, 15syl 17 . . 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 990 . . . . 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 991 . . . . . . 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 992 . . . . . . 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 993 . . . . . . 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 994 . . . . . . 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 995 . . . . . . 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 30376 . . . . . . 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 1205 . . . . . 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 962 . . . . . . 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 30376 . . . . . . 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 1205 . . . . . 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 30158 . . . . . 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 1187 . . . . 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 1079 . . . . 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 29579 . . . . 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 1187 . . . 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 28729 . . . 4  |-  ( ( ( [_ G  / 
g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A  ->  ( (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  B )
3432, 33syl 17 . . 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 29579 . . . . . 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 1187 . . . . 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 28729 . . . . 5  |-  ( (
[_ G  /  g ]_ X `  P )  e.  A  ->  ( [_ G  /  g ]_ X `  P )  e.  B )
3836, 37syl 17 . . . 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 29603 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ I  / 
g ]_ X  e.  T
)  ->  ( R `  [_ I  /  g ]_ X )  e.  B
)
4017, 27, 39syl2anc 645 . . . 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 14119 . . . 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 1187 . . 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 29579 . . . . . 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 1187 . . . . 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 28729 . . . . 5  |-  ( (
[_ I  /  g ]_ X `  P )  e.  A  ->  ( [_ I  /  g ]_ X `  P )  e.  B )
4644, 45syl 17 . . . 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 29603 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T
)  ->  ( R `  [_ G  /  g ]_ X )  e.  B
)
4817, 24, 47syl2anc 645 . . . 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 14119 . . . 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 1187 . . 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 14147 . . 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 1189 . 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 892 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   [_csb 3056   class class class wbr 3997    _I cid 4276   `'ccnv 4660    |` cres 4663    o. ccom 4665   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Latclat 14114   Atomscatm 28703   HLchlt 28790   LHypclh 29423   LTrncltrn 29540   trLctrl 29597
This theorem is referenced by:  cdlemk52  30393
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598
  Copyright terms: Public domain W3C validator