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Theorem dia2dimlem1 31179
Description: Lemma for dia2dim 31192. 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 446 . . . 4  |-  ( ph  ->  K  e.  HL )
4 dia2dimlem1.p . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
54simpld 446 . . . 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 30283 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( R `  F
)  e.  A )
14 dia2dimlem1.u . . . . 5  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
1514simpld 446 . . . 4  |-  ( ph  ->  U  e.  A )
166simpld 446 . . . . . 6  |-  ( ph  ->  F  e.  T )
177, 8, 9, 10ltrnel 30253 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1918simpld 446 . . . 4  |-  ( ph  ->  ( F `  P
)  e.  A )
20 dia2dimlem1.v . . . . 5  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
2120simpld 446 . . . 4  |-  ( ph  ->  V  e.  A )
224simprd 450 . . . . . 6  |-  ( ph  ->  -.  P  .<_  W )
237, 9, 10, 11trlle 30298 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
242, 16, 23syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( R `  F
)  .<_  W )
2514simprd 450 . . . . . . . 8  |-  ( ph  ->  U  .<_  W )
26 hllat 29478 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
273, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Lat )
28 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
2928, 8atbase 29404 . . . . . . . . . 10  |-  ( ( R `  F )  e.  A  ->  ( R `  F )  e.  ( Base `  K
) )
3013, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  e.  ( Base `  K ) )
3128, 8atbase 29404 . . . . . . . . . 10  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3215, 31syl 16 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( Base `  K ) )
332simprd 450 . . . . . . . . . 10  |-  ( ph  ->  W  e.  H )
3428, 9lhpbase 30112 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3533, 34syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
36 dia2dimlem1.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
3728, 7, 36latjle12 14418 . . . . . . . . 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 1186 . . . . . . . 8  |-  ( ph  ->  ( ( ( R `
 F )  .<_  W  /\  U  .<_  W )  <-> 
( ( R `  F )  .\/  U
)  .<_  W ) )
3924, 25, 38mpbi2and 888 . . . . . . 7  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  .<_  W )
4028, 8atbase 29404 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
415, 40syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
4228, 36, 8hlatjcl 29481 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
433, 13, 15, 42syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
4428, 7lattr 14412 . . . . . . . 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 1186 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  ( ( R `  F
)  .\/  U )  /\  ( ( R `  F )  .\/  U
)  .<_  W )  ->  P  .<_  W ) )
4639, 45mpan2d 656 . . . . . 6  |-  ( ph  ->  ( P  .<_  ( ( R `  F ) 
.\/  U )  ->  P  .<_  W ) )
4722, 46mtod 170 . . . . 5  |-  ( ph  ->  -.  P  .<_  ( ( R `  F ) 
.\/  U ) )
4820simprd 450 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
4918simprd 450 . . . . . . 7  |-  ( ph  ->  -.  ( F `  P )  .<_  W )
50 nbrne2 4171 . . . . . . 7  |-  ( ( V  .<_  W  /\  -.  ( F `  P
)  .<_  W )  ->  V  =/=  ( F `  P ) )
5148, 49, 50syl2anc 643 . . . . . 6  |-  ( ph  ->  V  =/=  ( F `
 P ) )
5251necomd 2633 . . . . 5  |-  ( ph  ->  ( F `  P
)  =/=  V )
5347, 52jca 519 . . . 4  |-  ( ph  ->  ( -.  P  .<_  ( ( R `  F
)  .\/  U )  /\  ( F `  P
)  =/=  V ) )
5427adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  K  e.  Lat )
5541adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  P  e.  ( Base `  K )
)
5628, 36, 8hlatjcl 29481 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  V  e.  A  /\  U  e.  A )  ->  ( V  .\/  U
)  e.  ( Base `  K ) )
573, 21, 15, 56syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( V  .\/  U
)  e.  ( Base `  K ) )
5857adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( V  .\/  U )  e.  (
Base `  K )
)
5935adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  W  e.  ( Base `  K )
)
607, 36, 8hlatlej2 29490 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
613, 19, 21, 60syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
6261adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  V  .<_  ( ( F `  P
)  .\/  V )
)
63 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)
6462, 63breqtrrd 4179 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  V  .<_  ( P  .\/  U ) )
65 dia2dimlem1.uv . . . . . . . . . . . 12  |-  ( ph  ->  U  =/=  V )
6665necomd 2633 . . . . . . . . . . 11  |-  ( ph  ->  V  =/=  U )
677, 36, 8hlatexch2 29510 . . . . . . . . . . 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 1197 . . . . . . . . . 10  |-  ( ph  ->  ( V  .<_  ( P 
.\/  U )  ->  P  .<_  ( V  .\/  U ) ) )
6968adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( V  .<_  ( P  .\/  U
)  ->  P  .<_  ( V  .\/  U ) ) )
7064, 69mpd 15 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  P  .<_  ( V  .\/  U ) )
7128, 8atbase 29404 . . . . . . . . . . . 12  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
7221, 71syl 16 . . . . . . . . . . 11  |-  ( ph  ->  V  e.  ( Base `  K ) )
7328, 7, 36latjle12 14418 . . . . . . . . . . 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 1186 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  .<_  W  /\  U  .<_  W )  <-> 
( V  .\/  U
)  .<_  W ) )
7548, 25, 74mpbi2and 888 . . . . . . . . 9  |-  ( ph  ->  ( V  .\/  U
)  .<_  W )
7675adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  ( V  .\/  U )  .<_  W )
7728, 7, 54, 55, 58, 59, 70, 76lattrd 14414 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  U )  =  ( ( F `  P
)  .\/  V )
)  ->  P  .<_  W )
7877ex 424 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  U )  =  ( ( F `  P ) 
.\/  V )  ->  P  .<_  W ) )
7978necon3bd 2587 . . . . 5  |-  ( ph  ->  ( -.  P  .<_  W  ->  ( P  .\/  U )  =/=  ( ( F `  P ) 
.\/  V ) ) )
8022, 79mpd 15 . . . 4  |-  ( ph  ->  ( P  .\/  U
)  =/=  ( ( F `  P ) 
.\/  V ) )
817, 36, 8hlatlej2 29490 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( F `  P
)  .<_  ( P  .\/  ( F `  P ) ) )
823, 5, 19, 81syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( F `  P
)  .<_  ( P  .\/  ( F `  P ) ) )
83 dia2dimlem1.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
847, 36, 83, 8, 9, 10, 11trlval2 30277 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8685oveq2d 6036 . . . . . . 7  |-  ( ph  ->  ( P  .\/  ( R `  F )
)  =  ( P 
.\/  ( ( P 
.\/  ( F `  P ) )  ./\  W ) ) )
8728, 36, 8hlatjcl 29481 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
883, 5, 19, 87syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
897, 36, 8hlatlej1 29489 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  ->  P  .<_  ( P  .\/  ( F `  P ) ) )
903, 5, 19, 89syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  P  .<_  ( P  .\/  ( F `  P
) ) )
9128, 7, 36, 83, 8atmod3i1 29978 . . . . . . . . 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 1197 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  (
( P  .\/  ( F `  P )
)  ./\  W )
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  ( P  .\/  W ) ) )
93 eqid 2387 . . . . . . . . . . . 12  |-  ( 1.
`  K )  =  ( 1. `  K
)
947, 36, 93, 8, 9lhpjat2 30135 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
952, 4, 94syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  W
)  =  ( 1.
`  K ) )
9695oveq2d 6036 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  ( F `  P ) )  ./\  ( 1. `  K ) ) )
97 hlol 29476 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  OL )
983, 97syl 16 . . . . . . . . . 10  |-  ( ph  ->  K  e.  OL )
9928, 83, 93olm11 29342 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
) )  ->  (
( P  .\/  ( F `  P )
)  ./\  ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
10098, 88, 99syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
10196, 100eqtrd 2419 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( P  .\/  W ) )  =  ( P  .\/  ( F `  P )
) )
10292, 101eqtrd 2419 . . . . . . 7  |-  ( ph  ->  ( P  .\/  (
( P  .\/  ( F `  P )
)  ./\  W )
)  =  ( P 
.\/  ( F `  P ) ) )
10386, 102eqtrd 2419 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( R `  F )
)  =  ( P 
.\/  ( F `  P ) ) )
10482, 103breqtrrd 4179 . . . . 5  |-  ( ph  ->  ( F `  P
)  .<_  ( P  .\/  ( R `  F ) ) )
105 dia2dimlem1.rf . . . . . . 7  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
10636, 8hlatjcom 29482 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
1073, 15, 21, 106syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
108105, 107breqtrd 4177 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
109 dia2dimlem1.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
1107, 36, 8hlatexch2 29510 . . . . . . 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 1197 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
112108, 111mpd 15 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
113104, 112jca 519 . . . 4  |-  ( ph  ->  ( ( F `  P )  .<_  ( P 
.\/  ( R `  F ) )  /\  V  .<_  ( ( R `
 F )  .\/  U ) ) )
1147, 36, 83, 8ps-2c 29642 . . . 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 1216 . . 3  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  e.  A )
1161, 115syl5eqel 2471 . 2  |-  ( ph  ->  Q  e.  A )
11728, 36, 8hlatjcl 29481 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
1183, 5, 15, 117syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
11928, 36, 8hlatjcl 29481 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
1203, 19, 21, 119syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
12128, 7, 83latmle1 14432 . . . . . . . . . . . 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 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  .<_  ( P  .\/  U ) )
1231, 122syl5eqbr 4186 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( P  .\/  U ) )
12428, 8atbase 29404 . . . . . . . . . . . . 13  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
125116, 124syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  ( Base `  K ) )
12628, 7, 83latlem12 14434 . . . . . . . . . . . 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 1186 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .<_  ( P  .\/  U )  /\  Q  .<_  W )  <-> 
Q  .<_  ( ( P 
.\/  U )  ./\  W ) ) )
128127biimpd 199 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .<_  ( P  .\/  U )  /\  Q  .<_  W )  ->  Q  .<_  ( ( P  .\/  U ) 
./\  W ) ) )
129123, 128mpand 657 . . . . . . . . 9  |-  ( ph  ->  ( Q  .<_  W  ->  Q  .<_  ( ( P 
.\/  U )  ./\  W ) ) )
130129imp 419 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  ( ( P  .\/  U
)  ./\  W )
)
131 eqid 2387 . . . . . . . . . . . . 13  |-  ( 0.
`  K )  =  ( 0. `  K
)
1327, 83, 131, 8, 9lhpmat 30144 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
1332, 4, 132syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( P  ./\  W
)  =  ( 0.
`  K ) )
134133oveq1d 6035 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  ./\  W )  .\/  U )  =  ( ( 0.
`  K )  .\/  U ) )
13528, 7, 36, 83, 8atmod4i1 29980 . . . . . . . . . . 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 1197 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  ./\  W )  .\/  U )  =  ( ( P 
.\/  U )  ./\  W ) )
13728, 36, 131olj02 29341 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  U  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  U
)  =  U )
13898, 32, 137syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0. `  K )  .\/  U
)  =  U )
139134, 136, 1383eqtr3d 2427 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  ./\  W )  =  U )
140139adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  ( ( P  .\/  U )  ./\  W )  =  U )
141130, 140breqtrd 4177 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  U )
142 hlatl 29475 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  AtLat )
1433, 142syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  AtLat )
144143adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  K  e.  AtLat
)
145116adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  e.  A )
14615adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  U  e.  A )
1477, 8atcmp 29426 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  U  e.  A )  ->  ( Q  .<_  U  <->  Q  =  U ) )
148144, 145, 146, 147syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  ( Q  .<_  U  <->  Q  =  U
) )
149141, 148mpbid 202 . . . . . 6  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  =  U )
15028, 7, 83latmle2 14433 . . . . . . . . . . . 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 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  .<_  ( ( F `  P )  .\/  V
) )
1521, 151syl5eqbr 4186 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( ( F `  P )  .\/  V ) )
15328, 7, 83latlem12 14434 . . . . . . . . . . . 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 1186 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .<_  ( ( F `  P
)  .\/  V )  /\  Q  .<_  W )  <-> 
Q  .<_  ( ( ( F `  P ) 
.\/  V )  ./\  W ) ) )
155154biimpd 199 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .<_  ( ( F `  P
)  .\/  V )  /\  Q  .<_  W )  ->  Q  .<_  ( ( ( F `  P
)  .\/  V )  ./\  W ) ) )
156152, 155mpand 657 . . . . . . . . 9  |-  ( ph  ->  ( Q  .<_  W  ->  Q  .<_  ( ( ( F `  P ) 
.\/  V )  ./\  W ) ) )
157156imp 419 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  ( ( ( F `  P )  .\/  V
)  ./\  W )
)
1587, 83, 131, 8, 9lhpmat 30144 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F `
 P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  -> 
( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
1592, 18, 158syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
160159oveq1d 6035 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( F `
 P )  ./\  W )  .\/  V )  =  ( ( 0.
`  K )  .\/  V ) )
16128, 8atbase 29404 . . . . . . . . . . . 12  |-  ( ( F `  P )  e.  A  ->  ( F `  P )  e.  ( Base `  K
) )
16219, 161syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  P
)  e.  ( Base `  K ) )
16328, 7, 36, 83, 8atmod4i1 29980 . . . . . . . . . . 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 1197 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( F `
 P )  ./\  W )  .\/  V )  =  ( ( ( F `  P ) 
.\/  V )  ./\  W ) )
16528, 36, 131olj02 29341 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  V  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  V
)  =  V )
16698, 72, 165syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0. `  K )  .\/  V
)  =  V )
167160, 164, 1663eqtr3d 2427 . . . . . . . . 9  |-  ( ph  ->  ( ( ( F `
 P )  .\/  V )  ./\  W )  =  V )
168167adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  ( (
( F `  P
)  .\/  V )  ./\  W )  =  V )
169157, 168breqtrd 4177 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  .<_  V )
17021adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Q  .<_  W )  ->  V  e.  A )
1717, 8atcmp 29426 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  V  e.  A )  ->  ( Q  .<_  V  <->  Q  =  V ) )
172144, 145, 170, 171syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  Q  .<_  W )  ->  ( Q  .<_  V  <->  Q  =  V
) )
173169, 172mpbid 202 . . . . . 6  |-  ( (
ph  /\  Q  .<_  W )  ->  Q  =  V )
174149, 173eqtr3d 2421 . . . . 5  |-  ( (
ph  /\  Q  .<_  W )  ->  U  =  V )
175174ex 424 . . . 4  |-  ( ph  ->  ( Q  .<_  W  ->  U  =  V )
)
176175necon3ad 2586 . . 3  |-  ( ph  ->  ( U  =/=  V  ->  -.  Q  .<_  W ) )
17765, 176mpd 15 . 2  |-  ( ph  ->  -.  Q  .<_  W )
178116, 177jca 519 1  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   0.cp0 14393   1.cp1 14394   Latclat 14401   OLcol 29289   Atomscatm 29378   AtLatcal 29379   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   trLctrl 30272
This theorem is referenced by:  dia2dimlem3  31181  dia2dimlem6  31184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273
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