Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1cvratex Unicode version

Theorem 1cvratex 30284
Description: There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)
Hypotheses
Ref Expression
1cvratex.b  |-  B  =  ( Base `  K
)
1cvratex.s  |-  .<  =  ( lt `  K )
1cvratex.u  |-  .1.  =  ( 1. `  K )
1cvratex.c  |-  C  =  (  <o  `  K )
1cvratex.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
1cvratex  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. p  e.  A  p  .<  X )
Distinct variable groups:    A, p    B, p    C, p    K, p    .< , p    .1. , p    X, p

Proof of Theorem 1cvratex
Dummy variables  q 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  K  e.  HL )
2 1cvratex.b . . . . 5  |-  B  =  ( Base `  K
)
3 1cvratex.u . . . . 5  |-  .1.  =  ( 1. `  K )
4 eqid 2296 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
5 1cvratex.c . . . . 5  |-  C  =  (  <o  `  K )
6 1cvratex.a . . . . 5  |-  A  =  ( Atoms `  K )
72, 3, 4, 5, 61cvrco 30283 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X C  .1.  <->  ( ( oc `  K
) `  X )  e.  A ) )
87biimp3a 1281 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  ( ( oc `  K ) `  X
)  e.  A )
9 eqid 2296 . . . 4  |-  ( join `  K )  =  (
join `  K )
109, 5, 62dim 30281 . . 3  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  X
)  e.  A )  ->  E. q  e.  A  E. r  e.  A  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )
111, 8, 10syl2anc 642 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. q  e.  A  E. r  e.  A  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )
12 simp11 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  K  e.  HL )
13 hlop 30174 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1412, 13syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  K  e.  OP )
15 hllat 30175 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
1612, 15syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  K  e.  Lat )
17 simp12 986 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  X  e.  B )
182, 4opoccl 30006 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1914, 17, 18syl2anc 642 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  X )  e.  B )
20 simp2l 981 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  q  e.  A )
212, 6atbase 30101 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
2220, 21syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  q  e.  B )
232, 9latjcl 14172 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  q  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B )
2416, 19, 22, 23syl3anc 1182 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) q )  e.  B )
252, 4opoccl 30006 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) )  e.  B )
2614, 24, 25syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  e.  B )
27 simp2r 982 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  r  e.  A )
282, 6atbase 30101 . . . . . . . . . . . . 13  |-  ( r  e.  A  ->  r  e.  B )
2927, 28syl 15 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  r  e.  B )
302, 9latjcl 14172 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B  /\  r  e.  B )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r )  e.  B )
3116, 24, 29, 30syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r )  e.  B )
322, 4opoccl 30006 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r )  e.  B )  -> 
( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) )  e.  B )
3314, 31, 32syl2anc 642 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) q ) (
join `  K )
r ) )  e.  B )
34 eqid 2296 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
35 eqid 2296 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
362, 34, 35op0le 29998 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) )  e.  B )  -> 
( 0. `  K
) ( le `  K ) ( ( oc `  K ) `
 ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )
3714, 33, 36syl2anc 642 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  ( 0. `  K ) ( le `  K ) ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) ) )
38 simp3r 984 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) )
39 1cvratex.s . . . . . . . . . . . 12  |-  .<  =  ( lt `  K )
402, 39, 5cvrlt 30082 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B  /\  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r )  e.  B )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) )  ->  ( (
( oc `  K
) `  X )
( join `  K )
q )  .<  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) )
4112, 24, 31, 38, 40syl31anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) q )  .< 
( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) )
422, 39, 4opltcon3b 30016 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B  /\  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r )  e.  B )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) 
.<  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r )  <-> 
( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) ) 
.<  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) ) )
4314, 24, 31, 42syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) 
.<  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r )  <-> 
( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) ) 
.<  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) ) )
4441, 43mpbid 201 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) q ) (
join `  K )
r ) )  .< 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) )
45 hlpos 30177 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Poset )
4612, 45syl 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  K  e.  Poset )
472, 35op0cl 29996 . . . . . . . . . . 11  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
4814, 47syl 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  ( 0. `  K )  e.  B )
492, 34, 39plelttr 14122 . . . . . . . . . 10  |-  ( ( K  e.  Poset  /\  (
( 0. `  K
)  e.  B  /\  ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) )  e.  B  /\  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  e.  B ) )  -> 
( ( ( 0.
`  K ) ( le `  K ) ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) )  /\  ( ( oc
`  K ) `  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) )  .<  ( ( oc `  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ) )  ->  ( 0. `  K )  .< 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) ) )
5046, 48, 33, 26, 49syl13anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( 0. `  K ) ( le
`  K ) ( ( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) q ) (
join `  K )
r ) )  /\  ( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) ) 
.<  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) )  ->  ( 0. `  K )  .<  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) ) ) )
5137, 44, 50mp2and 660 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  ( 0. `  K )  .< 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) )
5239pltne 14112 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( 0. `  K )  e.  B  /\  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  e.  B )  ->  (
( 0. `  K
)  .<  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) )  ->  ( 0. `  K )  =/=  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) ) ) )
5312, 48, 26, 52syl3anc 1182 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( 0. `  K
)  .<  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) )  ->  ( 0. `  K )  =/=  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) ) ) )
5451, 53mpd 14 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  ( 0. `  K )  =/=  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) )
5554necomd 2542 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  =/=  ( 0. `  K
) )
562, 34, 35, 6atle 30247 . . . . . 6  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) )  e.  B  /\  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  =/=  ( 0. `  K
) )  ->  E. p  e.  A  p ( le `  K ) ( ( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) ) )
5712, 26, 55, 56syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  E. p  e.  A  p ( le `  K ) ( ( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) ) )
58 simp3l 983 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) )
592, 39, 5cvrlt 30082 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B )  /\  ( ( oc `  K ) `  X
) C ( ( ( oc `  K
) `  X )
( join `  K )
q ) )  -> 
( ( oc `  K ) `  X
)  .<  ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) )
6012, 19, 24, 58, 59syl31anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  X )  .<  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) )
612, 39, 4opltcon3b 30016 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B )  -> 
( ( ( oc
`  K ) `  X )  .<  (
( ( oc `  K ) `  X
) ( join `  K
) q )  <->  ( ( oc `  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) )  .<  ( ( oc `  K ) `  ( ( oc `  K ) `  X
) ) ) )
6214, 19, 24, 61syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( ( oc `  K ) `  X
)  .<  ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  <->  ( ( oc `  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) )  .<  ( ( oc `  K ) `  ( ( oc `  K ) `  X
) ) ) )
6360, 62mpbid 201 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  .< 
( ( oc `  K ) `  (
( oc `  K
) `  X )
) )
642, 4opococ 30007 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
6514, 17, 64syl2anc 642 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  X ) )  =  X )
6663, 65breqtrd 4063 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  .<  X )
6766adantr 451 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  .<  X )
68 simpl11 1030 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  K  e.  HL )
6968, 45syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  K  e.  Poset )
702, 6atbase 30101 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
7170adantl 452 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  p  e.  B )
7226adantr 451 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  e.  B )
73 simpl12 1031 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  X  e.  B )
742, 34, 39plelttr 14122 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  (
p  e.  B  /\  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) )  e.  B  /\  X  e.  B ) )  -> 
( ( p ( le `  K ) ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) )  /\  ( ( oc
`  K ) `  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) )  .<  X )  ->  p  .<  X )
)
7569, 71, 72, 73, 74syl13anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  (
( p ( le
`  K ) ( ( oc `  K
) `  ( (
( oc `  K
) `  X )
( join `  K )
q ) )  /\  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) 
.<  X )  ->  p  .<  X ) )
7667, 75mpan2d 655 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) ) )  /\  p  e.  A )  ->  (
p ( le `  K ) ( ( oc `  K ) `
 ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) )  ->  p  .<  X ) )
7776reximdva 2668 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  ( E. p  e.  A  p ( le `  K ) ( ( oc `  K ) `
 ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) )  ->  E. p  e.  A  p  .<  X ) )
7857, 77mpd 14 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  /\  ( q  e.  A  /\  r  e.  A
)  /\  ( (
( oc `  K
) `  X ) C ( ( ( oc `  K ) `
 X ) (
join `  K )
q )  /\  (
( ( oc `  K ) `  X
) ( join `  K
) q ) C ( ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) ( join `  K ) r ) ) )  ->  E. p  e.  A  p  .<  X )
79783exp 1150 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  ( ( q  e.  A  /\  r  e.  A )  ->  (
( ( ( oc
`  K ) `  X ) C ( ( ( oc `  K ) `  X
) ( join `  K
) q )  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) C ( ( ( ( oc `  K
) `  X )
( join `  K )
q ) ( join `  K ) r ) )  ->  E. p  e.  A  p  .<  X ) ) )
8079rexlimdvv 2686 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  ( E. q  e.  A  E. r  e.  A  ( ( ( oc `  K ) `
 X ) C ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  /\  ( ( ( oc `  K ) `
 X ) (
join `  K )
q ) C ( ( ( ( oc
`  K ) `  X ) ( join `  K ) q ) ( join `  K
) r ) )  ->  E. p  e.  A  p  .<  X ) )
8111, 80mpd 14 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. p  e.  A  p  .<  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   Posetcpo 14090   ltcplt 14091   joincjn 14094   0.cp0 14159   1.cp1 14160   Latclat 14167   OPcops 29984    <o ccvr 30074   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  1cvratlt  30285  lhpexlt  30813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
  Copyright terms: Public domain W3C validator