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Theorem 1cvratex 32963
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 1006 . . 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 2423 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
5 1cvratex.c . . . . 5  |-  C  =  (  <o  `  K )
6 1cvratex.a . . . . 5  |-  A  =  ( Atoms `  K )
72, 3, 4, 5, 61cvrco 32962 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X C  .1.  <->  ( ( oc `  K
) `  X )  e.  A ) )
87biimp3a 1365 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  ( ( oc `  K ) `  X
)  e.  A )
9 eqid 2423 . . . 4  |-  ( join `  K )  =  (
join `  K )
109, 5, 62dim 32960 . . 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 666 . 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 1036 . . . . . 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 32853 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1412, 13syl 17 . . . . . . 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 32854 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
1612, 15syl 17 . . . . . . . 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 1037 . . . . . . . . 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 32685 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1914, 17, 18syl2anc 666 . . . . . . . 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 1032 . . . . . . . . 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 32780 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
2220, 21syl 17 . . . . . . . 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 16290 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  q  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B )
2416, 19, 22, 23syl3anc 1265 . . . . . . 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 32685 . . . . . . 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 666 . . . . . 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 1033 . . . . . . . . . . . . 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 32780 . . . . . . . . . . . . 13  |-  ( r  e.  A  ->  r  e.  B )
2927, 28syl 17 . . . . . . . . . . . 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 16290 . . . . . . . . . . . 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 1265 . . . . . . . . . . 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 32685 . . . . . . . . . . 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 666 . . . . . . . . . 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 2423 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
35 eqid 2423 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
362, 34, 35op0le 32677 . . . . . . . . . 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 666 . . . . . . . . 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 1035 . . . . . . . . . . 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 32761 . . . . . . . . . . 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 1268 . . . . . . . . . 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 32695 . . . . . . . . . . 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 1265 . . . . . . . . . 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 214 . . . . . . . . 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 32856 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Poset )
4612, 45syl 17 . . . . . . . . . 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 32675 . . . . . . . . . . 11  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
4814, 47syl 17 . . . . . . . . . 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 16211 . . . . . . . . . 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 1267 . . . . . . . . 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 684 . . . . . . . 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 16201 . . . . . . . . 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 1265 . . . . . . . 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 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 ) ) )  ->  ( 0. `  K )  =/=  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) q ) ) )
5554necomd 2696 . . . . . 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 32926 . . . . . 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 1265 . . . . 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 1034 . . . . . . . . . . 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 32761 . . . . . . . . . . 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 1268 . . . . . . . . . 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 32695 . . . . . . . . . . 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 1265 . . . . . . . . . 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 214 . . . . . . . . 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 32686 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
6514, 17, 64syl2anc 666 . . . . . . . . 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 4446 . . . . . . . 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 467 . . . . . . 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 1081 . . . . . . . . 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 17 . . . . . . . 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 32780 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
7170adantl 468 . . . . . . . 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 467 . . . . . . . 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 1082 . . . . . . . 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 16211 . . . . . . . 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 1267 . . . . . . 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 679 . . . . . 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 2901 . . . . 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 15 . . . 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 1205 . . 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 2924 . 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 15 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. p  e.  A  p  .<  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Basecbs 15114   lecple 15190   occoc 15191   Posetcpo 16178   ltcplt 16179   joincjn 16182   0.cp0 16276   1.cp1 16277   Latclat 16284   OPcops 32663    <o ccvr 32753   Atomscatm 32754   HLchlt 32841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842
This theorem is referenced by:  1cvratlt  32964  lhpexlt  33492
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