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Theorem cdlemg42 36871
Description: Part of proof of Lemma G of [Crawley] p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
Hypotheses
Ref Expression
cdlemg42.l  |-  .<_  =  ( le `  K )
cdlemg42.j  |-  .\/  =  ( join `  K )
cdlemg42.a  |-  A  =  ( Atoms `  K )
cdlemg42.h  |-  H  =  ( LHyp `  K
)
cdlemg42.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg42.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg42  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  -.  ( G `  P )  .<_  ( P  .\/  ( F `  P )
) )

Proof of Theorem cdlemg42
StepHypRef Expression
1 simp33 1032 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( R `  F )  =/=  ( R `  G )
)
2 simpl1l 1045 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  K  e.  HL )
3 simp31l 1117 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  P  e.  A )
43adantr 463 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  P  e.  A )
5 simp1 994 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simp2l 1020 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  F  e.  T )
7 cdlemg42.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
8 cdlemg42.a . . . . . . . . . . . 12  |-  A  =  ( Atoms `  K )
9 cdlemg42.h . . . . . . . . . . . 12  |-  H  =  ( LHyp `  K
)
10 cdlemg42.t . . . . . . . . . . . 12  |-  T  =  ( ( LTrn `  K
) `  W )
117, 8, 9, 10ltrnat 36280 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
125, 6, 3, 11syl3anc 1226 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  P )  e.  A
)
1312adantr 463 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( F `  P )  e.  A )
14 cdlemg42.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
157, 14, 8hlatlej1 35515 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  ->  P  .<_  ( P  .\/  ( F `  P ) ) )
162, 4, 13, 15syl3anc 1226 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  P  .<_  ( P  .\/  ( F `  P )
) )
17 simpr 459 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( G `  P )  .<_  ( P  .\/  ( F `  P )
) )
18 hllat 35504 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
192, 18syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  K  e.  Lat )
20 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
2120, 8atbase 35430 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
224, 21syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  P  e.  ( Base `  K
) )
23 simp2r 1021 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  G  e.  T )
247, 8, 9, 10ltrnat 36280 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
255, 23, 3, 24syl3anc 1226 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( G `  P )  e.  A
)
2625adantr 463 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( G `  P )  e.  A )
2720, 8atbase 35430 . . . . . . . . . 10  |-  ( ( G `  P )  e.  A  ->  ( G `  P )  e.  ( Base `  K
) )
2826, 27syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( G `  P )  e.  ( Base `  K
) )
2920, 14, 8hlatjcl 35507 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
302, 4, 13, 29syl3anc 1226 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
3120, 7, 14latjle12 15894 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( G `  P )  e.  ( Base `  K
)  /\  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
) )  ->  (
( P  .<_  ( P 
.\/  ( F `  P ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  <->  ( P  .\/  ( G `  P
) )  .<_  ( P 
.\/  ( F `  P ) ) ) )
3219, 22, 28, 30, 31syl13anc 1228 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  (
( P  .<_  ( P 
.\/  ( F `  P ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  <->  ( P  .\/  ( G `  P
) )  .<_  ( P 
.\/  ( F `  P ) ) ) )
3316, 17, 32mpbi2and 919 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( P  .\/  ( G `  P ) )  .<_  ( P  .\/  ( F `
 P ) ) )
34 simpl32 1076 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( G `  P )  =/=  P )
3534necomd 2725 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  P  =/=  ( G `  P
) )
367, 14, 8ps-1 35617 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( G `  P
)  e.  A  /\  P  =/=  ( G `  P ) )  /\  ( P  e.  A  /\  ( F `  P
)  e.  A ) )  ->  ( ( P  .\/  ( G `  P ) )  .<_  ( P  .\/  ( F `
 P ) )  <-> 
( P  .\/  ( G `  P )
)  =  ( P 
.\/  ( F `  P ) ) ) )
372, 4, 26, 35, 4, 13, 36syl132anc 1244 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  (
( P  .\/  ( G `  P )
)  .<_  ( P  .\/  ( F `  P ) )  <->  ( P  .\/  ( G `  P ) )  =  ( P 
.\/  ( F `  P ) ) ) )
3833, 37mpbid 210 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( P  .\/  ( G `  P ) )  =  ( P  .\/  ( F `  P )
) )
3938oveq1d 6285 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  (
( P  .\/  ( G `  P )
) ( meet `  K
) W )  =  ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W ) )
40 simpl1 997 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
41 simpl2r 1048 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  G  e.  T )
42 simpl31 1075 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
43 eqid 2454 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
44 cdlemg42.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
457, 14, 43, 8, 9, 10, 44trlval2 36304 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
) ( meet `  K
) W ) )
4640, 41, 42, 45syl3anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( R `  G )  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
47 simpl2l 1047 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  F  e.  T )
487, 14, 43, 8, 9, 10, 44trlval2 36304 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) W ) )
4940, 47, 42, 48syl3anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( R `  F )  =  ( ( P 
.\/  ( F `  P ) ) (
meet `  K ) W ) )
5039, 46, 493eqtr4rd 2506 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `
 F )  =/=  ( R `  G
) ) )  /\  ( G `  P ) 
.<_  ( P  .\/  ( F `  P )
) )  ->  ( R `  F )  =  ( R `  G ) )
5150ex 432 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( ( G `  P )  .<_  ( P  .\/  ( F `  P )
)  ->  ( R `  F )  =  ( R `  G ) ) )
5251necon3ad 2664 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( ( R `  F )  =/=  ( R `  G
)  ->  -.  ( G `  P )  .<_  ( P  .\/  ( F `  P )
) ) )
531, 52mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  -.  ( G `  P )  .<_  ( P  .\/  ( F `  P )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Latclat 15877   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300
This theorem is referenced by:  cdlemg43  36872
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