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Theorem 1cvratex 34486
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 996 . . 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 2467 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
5 1cvratex.c . . . . 5  |-  C  =  (  <o  `  K )
6 1cvratex.a . . . . 5  |-  A  =  ( Atoms `  K )
72, 3, 4, 5, 61cvrco 34485 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X C  .1.  <->  ( ( oc `  K
) `  X )  e.  A ) )
87biimp3a 1328 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  ( ( oc `  K ) `  X
)  e.  A )
9 eqid 2467 . . . 4  |-  ( join `  K )  =  (
join `  K )
109, 5, 62dim 34483 . . 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 661 . 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 1026 . . . . . 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 34376 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1412, 13syl 16 . . . . . . 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 34377 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
1612, 15syl 16 . . . . . . . 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 1027 . . . . . . . . 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 34208 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
1914, 17, 18syl2anc 661 . . . . . . . 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 1022 . . . . . . . . 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 34303 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
2220, 21syl 16 . . . . . . . 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 15541 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  q  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) q )  e.  B )
2416, 19, 22, 23syl3anc 1228 . . . . . . 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 34208 . . . . . . 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 661 . . . . . 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 1023 . . . . . . . . . . . . 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 34303 . . . . . . . . . . . . 13  |-  ( r  e.  A  ->  r  e.  B )
2927, 28syl 16 . . . . . . . . . . . 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 15541 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 34208 . . . . . . . . . . 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 661 . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
35 eqid 2467 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
362, 34, 35op0le 34200 . . . . . . . . . 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 661 . . . . . . . . 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 1025 . . . . . . . . . . 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 34284 . . . . . . . . . . 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 1231 . . . . . . . . . 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 34218 . . . . . . . . . . 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 1228 . . . . . . . . . 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 210 . . . . . . . . 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 34379 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Poset )
4612, 45syl 16 . . . . . . . . . 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 34198 . . . . . . . . . . 11  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
4814, 47syl 16 . . . . . . . . . 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 15462 . . . . . . . . . 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 1230 . . . . . . . . 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 679 . . . . . . . 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 15452 . . . . . . . . 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 1228 . . . . . . . 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 2738 . . . . . 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 34449 . . . . . 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 1228 . . . . 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 1024 . . . . . . . . . . 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 34284 . . . . . . . . . . 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 1231 . . . . . . . . . 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 34218 . . . . . . . . . . 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 1228 . . . . . . . . . 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 210 . . . . . . . . 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 34209 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
6514, 17, 64syl2anc 661 . . . . . . . . 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 4471 . . . . . . . 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 465 . . . . . . 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 1071 . . . . . . . . 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 16 . . . . . . . 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 34303 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
7170adantl 466 . . . . . . . 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 465 . . . . . . . 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 1072 . . . . . . . 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 15462 . . . . . . . 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 1230 . . . . . . 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 674 . . . . . 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 2938 . . . . 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 1195 . . 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 2961 . 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   occoc 14566   Posetcpo 15430   ltcplt 15431   joincjn 15434   0.cp0 15527   1.cp1 15528   Latclat 15535   OPcops 34186    <o ccvr 34276   Atomscatm 34277   HLchlt 34364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365
This theorem is referenced by:  1cvratlt  34487  lhpexlt  35015
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