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Theorem dia2dimlem1 31254
Description: Lemma for dia2dim 31267. Show properties of the auxiliary atom  Q. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem1.l  |-  .<_  =  ( le `  K )
dia2dimlem1.j  |-  .\/  =  ( join `  K )
dia2dimlem1.m  |-  ./\  =  ( meet `  K )
dia2dimlem1.a  |-  A  =  ( Atoms `  K )
dia2dimlem1.h  |-  H  =  ( LHyp `  K
)
dia2dimlem1.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem1.r  |-  R  =  ( ( trL `  K
) `  W )
dia2dimlem1.q  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
dia2dimlem1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem1.u  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
dia2dimlem1.v  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
dia2dimlem1.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem1.f  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
dia2dimlem1.rf  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
dia2dimlem1.uv  |-  ( ph  ->  U  =/=  V )
dia2dimlem1.ru  |-  ( ph  ->  ( R `  F
)  =/=  U )
Assertion
Ref Expression
dia2dimlem1  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )

Proof of Theorem dia2dimlem1
StepHypRef Expression
1 dia2dimlem1.q . . 3  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
2 dia2dimlem1.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
32simpld 445 . . . 4  |-  ( ph  ->  K  e.  HL )
4 dia2dimlem1.p . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
54simpld 445 . . . 4  |-  ( ph  ->  P  e.  A )
6 dia2dimlem1.f . . . . 5  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
7 dia2dimlem1.l . . . . . 6  |-  .<_  =  ( le `  K )
8 dia2dimlem1.a . . . . . 6  |-  A  =  ( Atoms `  K )
9 dia2dimlem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
10 dia2dimlem1.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
11 dia2dimlem1.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
127, 8, 9, 10, 11trlat 30358 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
132, 4, 6, 12syl3anc 1182 . . . 4  |-  ( ph  ->  ( R `  F
)  e.  A )
14 dia2dimlem1.u . . . . 5  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
1514simpld 445 . . . 4  |-  ( ph  ->  U  e.  A )
166simpld 445 . . . . . 6  |-  ( ph  ->  F  e.  T )
177, 8, 9, 10ltrnel 30328 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
182, 16, 4, 17syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1918simpld 445 . . . 4  |-  ( ph  ->  ( F `  P
)  e.  A )
20 dia2dimlem1.v . . . . 5  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
2120simpld 445 . . . 4  |-  ( ph  ->  V  e.  A )
224simprd 449 . . . . . 6  |-  ( ph  ->  -.  P  .<_  W )
237, 9, 10, 11trlle 30373 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
242, 16, 23syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( R `  F
)  .<_  W )
2514simprd 449 . . . . . . . 8  |-  ( ph  ->  U  .<_  W )
26 hllat 29553 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
273, 26syl 15 . . . . . . . . 9  |-  ( ph  ->  K  e.  Lat )
28 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
2928, 8atbase 29479 . . . . . . . . . 10  |-  ( ( R `  F )  e.  A  ->  ( R `  F )  e.  ( Base `  K
) )
3013, 29syl 15 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  e.  ( Base `  K ) )
3128, 8atbase 29479 . . . . . . . . . 10  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3215, 31syl 15 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( Base `  K ) )
332simprd 449 . . . . . . . . . 10  |-  ( ph  ->  W  e.  H )
3428, 9lhpbase 30187 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3533, 34syl 15 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
36 dia2dimlem1.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
3728, 7, 36latjle12 14168 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( R `  F )  e.  (
Base `  K )  /\  U  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( ( ( R `  F ) 
.<_  W  /\  U  .<_  W )  <->  ( ( R `
 F )  .\/  U )  .<_  W )
)
3827, 30, 32, 35, 37syl13anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( ( R `
 F )  .<_  W  /\  U  .<_  W )  <-> 
( ( R `  F )  .\/  U
)  .<_  W ) )
3924, 25, 38mpbi2and 887 . . . . . . 7  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  .<_  W )
4028, 8atbase 29479 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
415, 40syl 15 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
4228, 36, 8hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
433, 13, 15, 42syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
4428, 7lattr 14162 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( ( R `  F )  .\/  U
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( ( R `  F
)  .\/  U )  /\  ( ( R `  F )  .\/  U
)  .<_  W )  ->  P  .<_  W ) )
4527, 41, 43, 35, 44syl13anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  ( ( R `  F
)  .\/  U )  /\  ( ( R `  F )  .\/  U
)  .<_  W )  ->  P  .<_  W ) )
4639, 45mpan2d 655 . . . . . 6  |-  ( ph  ->  ( P  .<_  ( ( R `  F ) 
.\/  U )  ->  P  .<_  W ) )
4722, 46mtod 168 . . . . 5  |-  ( ph  ->  -.  P  .<_  ( ( R `  F ) 
.\/  U ) )
4820simprd 449 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
4918simprd 449 . . . . . . 7  |-  ( ph  ->  -.  ( F `  P )  .<_  W )
50 nbrne2 4041 . . . . . . 7  |-  ( ( V  .<_  W  /\  -.  ( F `  P
)  .<_  W )  ->  V  =/=  ( F `  P ) )
5148, 49, 50syl2anc 642 . . . . . 6  |-  ( ph  ->  V  =/=  ( F `
 P ) )
5251necomd 2529 . . . . 5  |-  ( ph  ->  ( F `  P
)  =/=  V )
5347, 52jca 518 . . . 4  |-  ( ph  ->  ( -.  P  .<_  ( ( R `  F
)  .\/  U )  /\  ( F `  P
)  =/=  V ) )
5427adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  K  e.  Lat )
5541adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  P  e.  ( Base `  K )
)
5628, 36, 8hlatjcl 29556 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  V  e.  A  /\  U  e.  A )  ->  ( V  .\/  U
)  e.  ( Base `  K ) )
573, 21, 15, 56syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( V  .\/  U
)  e.  ( Base `  K ) )
5857adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( V  .\/  U )  e.  (
Base `  K )
)
5935adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  W  e.  ( Base `  K )
)
607, 36, 8hlatlej2 29565 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
613, 19, 21, 60syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
6261adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  V  .<_  ( ( F `  P
)  .\/  V )
)
63 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)
6462, 63breqtrrd 4049 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  V  .<_  ( P  .\/  U ) )
65 dia2dimlem1.uv . . . . . . . . . . . 12  |-  ( ph  ->  U  =/=  V )
6665necomd 2529 . . . . . . . . . . 11  |-  ( ph  ->  V  =/=  U )
677, 36, 8hlatexch2 29585 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( V  e.  A  /\  P  e.  A  /\  U  e.  A
)  /\  V  =/=  U )  ->  ( V  .<_  ( P  .\/  U
)  ->  P  .<_  ( V  .\/  U ) ) )
683, 21, 5, 15, 66, 67syl131anc 1195 . . . . . . . . . 10  |-  ( ph  ->  ( V  .<_  ( P 
.\/  U )  ->  P  .<_  ( V  .\/  U ) ) )
6968adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( V  .<_  ( P  .\/  U
)  ->  P  .<_  ( V  .\/  U ) ) )
7064, 69mpd 14 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  P  .<_  ( V  .\/  U ) )
7128, 8atbase 29479 . . . . . . . . . . . 12  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
7221, 71syl 15 . . . . . . . . . . 11  |-  ( ph  ->  V  e.  ( Base `  K ) )
7328, 7, 36latjle12 14168 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( V  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( V  .<_  W  /\  U  .<_  W )  <-> 
( V  .\/  U
)  .<_  W ) )
7427, 72, 32, 35, 73syl13anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  .<_  W  /\  U  .<_  W )  <-> 
( V  .\/  U
)  .<_  W ) )
7548, 25, 74mpbi2and 887 . . . . . . . . 9  |-  ( ph  ->  ( V  .\/  U
)  .<_  W )
7675adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( V  .\/  U )  .<_  W )
7728, 7, 54, 55, 58, 59, 70, 76lattrd 14164 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  P  .<_  W )
7877ex 423 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  U )  =  ( ( F `  P ) 
.\/  V )  ->  P  .<_  W ) )
7978necon3bd 2483 . . . . 5  |-  ( ph  ->  ( -.  P  .<_  W  ->  ( P  .\/  U )  =/=  ( ( F `  P ) 
.\/  V ) ) )
8022, 79mpd 14 . . . 4  |-  ( ph  ->  ( P  .\/  U
)  =/=  ( ( F `  P ) 
.\/  V ) )
817, 36, 8hlatlej2 29565 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( F `  P
)  .<_  ( P  .\/  ( F `  P ) ) )
823, 5, 19, 81syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( F `  P
)  .<_  ( P  .\/  ( F `  P ) ) )
83 dia2dimlem1.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
847, 36, 83, 8, 9, 10, 11trlval2 30352 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
852, 16, 4, 84syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8685oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( P  .\/  ( R `  F )
)  =  ( P 
.\/  ( ( P 
.\/  ( F `  P ) )  ./\  W ) ) )
8728, 36, 8hlatjcl 29556 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
883, 5, 19, 87syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
897, 36, 8hlatlej1 29564 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  ->  P  .<_  ( P  .\/  ( F `  P ) ) )
903, 5, 19, 89syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  P  .<_  ( P  .\/  ( F `  P
) ) )
9128, 7, 36, 83, 8atmod3i1 30053 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  ( F `  P )
) )  ->  ( P  .\/  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )  =  ( ( P  .\/  ( F `  P )
)  ./\  ( P  .\/  W ) ) )
923, 5, 88, 35, 90, 91syl131anc 1195 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  (
( P  .\/  ( F `  P )
)  ./\  W )
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  ( P  .\/  W ) ) )
93 eqid 2283 . . . . . . . . . . . 12  |-  ( 1.
`  K )  =  ( 1. `  K
)
947, 36, 93, 8, 9lhpjat2 30210 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
952, 4, 94syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  W
)  =  ( 1.
`  K ) )
9695oveq2d 5874 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  ( F `  P ) )  ./\  ( 1. `  K ) ) )
97 hlol 29551 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  OL )
983, 97syl 15 . . . . . . . . . 10  |-  ( ph  ->  K  e.  OL )
9928, 83, 93olm11 29417 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
) )  ->  (
( P  .\/  ( F `  P )
)  ./\  ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
10098, 88, 99syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
10196, 100eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( P  .\/  W ) )  =  ( P  .\/  ( F `  P )
) )
10292, 101eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( P  .\/  (
( P  .\/  ( F `  P )
)  ./\  W )
)  =  ( P 
.\/  ( F `  P ) ) )
10386, 102eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( R `  F )
)  =  ( P 
.\/  ( F `  P ) ) )
10482, 103breqtrrd 4049 . . . . 5  |-  ( ph  ->  ( F `  P
)  .<_  ( P  .\/  ( R `  F ) ) )
105 dia2dimlem1.rf . . . . . . 7  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
10636, 8hlatjcom 29557 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
1073, 15, 21, 106syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
108105, 107breqtrd 4047 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
109 dia2dimlem1.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
1107, 36, 8hlatexch2 29585 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( R `  F )  e.  A  /\  V  e.  A  /\  U  e.  A
)  /\  ( R `  F )  =/=  U
)  ->  ( ( R `  F )  .<_  ( V  .\/  U
)  ->  V  .<_  ( ( R `  F
)  .\/  U )
) )
1113, 13, 21, 15, 109, 110syl131anc 1195 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
112108, 111mpd 14 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
113104, 112jca 518 . . . 4  |-  ( ph  ->  ( ( F `  P )  .<_  ( P 
.\/  ( R `  F ) )  /\  V  .<_  ( ( R `
 F )  .\/  U ) ) )
1147, 36, 83, 8ps-2c 29717 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  ( R `  F )  e.  A )  /\  ( U  e.  A  /\  ( F `  P
)  e.  A  /\  V  e.  A )  /\  ( ( -.  P  .<_  ( ( R `  F )  .\/  U
)  /\  ( F `  P )  =/=  V
)  /\  ( P  .\/  U )  =/=  (
( F `  P
)  .\/  V )  /\  ( ( F `  P )  .<_  ( P 
.\/  ( R `  F ) )  /\  V  .<_  ( ( R `
 F )  .\/  U ) ) ) )  ->  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  e.  A
)
1153, 5, 13, 15, 19, 21, 53, 80, 113, 114syl333anc 1214 . . 3  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  e.  A )
1161, 115syl5eqel 2367 . 2  |-  ( ph  ->  Q  e.  A )
11728, 36, 8hlatjcl 29556 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
1183, 5, 15, 117syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
11928, 36, 8hlatjcl 29556 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
1203, 19, 21, 119syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
12128, 7, 83latmle1 14182 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  U )  e.  ( Base `  K
)  /\  ( ( F `  P )  .\/  V )  e.  (
Base `  K )
)  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .<_  ( P 
.\/  U ) )
12227, 118, 120, 121syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  .<_  ( P  .\/  U ) )
1231, 122syl5eqbr 4056 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( P  .\/  U ) )
12428, 8atbase 29479 . . . . . . . . . . . . 13  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
125116, 124syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  ( Base `  K ) )
12628, 7, 83latlem12 14184 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( Q  .<_  ( P 
.\/  U )  /\  Q  .<_  W )  <->  Q  .<_  ( ( P  .\/  U
)  ./\  W )
) )
12727, 125, 118, 35, 126syl13anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .<_  ( P  .\/  U )  /\  Q  .<_  W )  <-> 
Q  .<_  ( ( P 
.\/  U )  ./\  W ) ) )
128127biimpd 198 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .<_  ( P  .\/  U )  /\  Q  .<_  W )  ->  Q  .<_  ( ( P  .\/  U ) 
./\  W ) ) )
129123, 128mpand 656 . . . . . . . . 9  |-  ( ph  ->  ( Q  .<_  W  ->  Q  .<_  ( ( P 
.\/  U )  ./\  W ) ) )
130129imp 418 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  ( ( P  .\/  U
)  ./\  W )
)
131 eqid 2283 . . . . . . . . . . . . 13  |-  ( 0.
`  K )  =  ( 0. `  K
)
1327, 83, 131, 8, 9lhpmat 30219 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
1332, 4, 132syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( P  ./\  W
)  =  ( 0.
`  K ) )
134133oveq1d 5873 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  ./\  W )  .\/  U )  =  ( ( 0.
`  K )  .\/  U ) )
13528, 7, 36, 83, 8atmod4i1 30055 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  P  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) )  /\  U  .<_  W )  -> 
( ( P  ./\  W )  .\/  U )  =  ( ( P 
.\/  U )  ./\  W ) )
1363, 15, 41, 35, 25, 135syl131anc 1195 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  ./\  W )  .\/  U )  =  ( ( P 
.\/  U )  ./\  W ) )
13728, 36, 131olj02 29416 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  U  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  U
)  =  U )
13898, 32, 137syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0. `  K )  .\/  U
)  =  U )
139134, 136, 1383eqtr3d 2323 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  ./\  W )  =  U )
140139adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  ( ( P  .\/  U )  ./\  W )  =  U )
141130, 140breqtrd 4047 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  U )
142 hlatl 29550 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  AtLat )
1433, 142syl 15 . . . . . . . . 9  |-  ( ph  ->  K  e.  AtLat )
144143adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  K  e.  AtLat
)
145116adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  e.  A )
14615adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  U  e.  A )
1477, 8atcmp 29501 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  U  e.  A )  ->  ( Q  .<_  U  <->  Q  =  U ) )
148144, 145, 146, 147syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  ( Q  .<_  U  <->  Q  =  U
) )
149141, 148mpbid 201 . . . . . 6  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  =  U )
15028, 7, 83latmle2 14183 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  U )  e.  ( Base `  K
)  /\  ( ( F `  P )  .\/  V )  e.  (
Base `  K )
)  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .<_  ( ( F `  P ) 
.\/  V ) )
15127, 118, 120, 150syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  .<_  ( ( F `  P )  .\/  V
) )
1521, 151syl5eqbr 4056 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( ( F `  P )  .\/  V ) )
15328, 7, 83latlem12 14184 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( ( F `  P )  .\/  V
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( Q  .<_  ( ( F `  P
)  .\/  V )  /\  Q  .<_  W )  <-> 
Q  .<_  ( ( ( F `  P ) 
.\/  V )  ./\  W ) ) )
15427, 125, 120, 35, 153syl13anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .<_  ( ( F `  P
)  .\/  V )  /\  Q  .<_  W )  <-> 
Q  .<_  ( ( ( F `  P ) 
.\/  V )  ./\  W ) ) )
155154biimpd 198 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .<_  ( ( F `  P
)  .\/  V )  /\  Q  .<_  W )  ->  Q  .<_  ( ( ( F `  P
)  .\/  V )  ./\  W ) ) )
156152, 155mpand 656 . . . . . . . . 9  |-  ( ph  ->  ( Q  .<_  W  ->  Q  .<_  ( ( ( F `  P ) 
.\/  V )  ./\  W ) ) )
157156imp 418 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  ( ( ( F `  P )  .\/  V
)  ./\  W )
)
1587, 83, 131, 8, 9lhpmat 30219 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F `
 P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  -> 
( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
1592, 18, 158syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
160159oveq1d 5873 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( F `
 P )  ./\  W )  .\/  V )  =  ( ( 0.
`  K )  .\/  V ) )
16128, 8atbase 29479 . . . . . . . . . . . 12  |-  ( ( F `  P )  e.  A  ->  ( F `  P )  e.  ( Base `  K
) )
16219, 161syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  P
)  e.  ( Base `  K ) )
16328, 7, 36, 83, 8atmod4i1 30055 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( V  e.  A  /\  ( F `  P
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  V  .<_  W )  ->  (
( ( F `  P )  ./\  W
)  .\/  V )  =  ( ( ( F `  P ) 
.\/  V )  ./\  W ) )
1643, 21, 162, 35, 48, 163syl131anc 1195 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( F `
 P )  ./\  W )  .\/  V )  =  ( ( ( F `  P ) 
.\/  V )  ./\  W ) )
16528, 36, 131olj02 29416 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  V  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  V
)  =  V )
16698, 72, 165syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0. `  K )  .\/  V
)  =  V )
167160, 164, 1663eqtr3d 2323 . . . . . . . . 9  |-  ( ph  ->  ( ( ( F `
 P )  .\/  V )  ./\  W )  =  V )
168167adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  ( (
( F `  P
)  .\/  V )  ./\  W )  =  V )
169157, 168breqtrd 4047 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  V )
17021adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  V  e.  A )
1717, 8atcmp 29501 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  V  e.  A )  ->  ( Q  .<_  V  <->  Q  =  V ) )
172144, 145, 170, 171syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  ( Q  .<_  V  <->  Q  =  V
) )
173169, 172mpbid 201 . . . . . 6  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  =  V )
174149, 173eqtr3d 2317 . . . . 5  |-  ( (
ph  /\  Q  .<_  W )  ->  U  =  V )
175174ex 423 . . . 4  |-  ( ph  ->  ( Q  .<_  W  ->  U  =  V )
)
176175necon3ad 2482 . . 3  |-  ( ph  ->  ( U  =/=  V  ->  -.  Q  .<_  W ) )
17765, 176mpd 14 . 2  |-  ( ph  ->  -.  Q  .<_  W )
178116, 177jca 518 1  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   1.cp1 14144   Latclat 14151   OLcol 29364   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  dia2dimlem3  31256  dia2dimlem6  31259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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