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Theorem 1cvratex 35340
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 2457 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
5 1cvratex.c . . . . 5  |-  C  =  (  <o  `  K )
6 1cvratex.a . . . . 5  |-  A  =  ( Atoms `  K )
72, 3, 4, 5, 61cvrco 35339 . . . 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 2457 . . . 4  |-  ( join `  K )  =  (
join `  K )
109, 5, 62dim 35337 . . 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 35230 . . . . . . . 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 35231 . . . . . . . . 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 35062 . . . . . . . . 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 35157 . . . . . . . . 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 15808 . . . . . . . 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 35062 . . . . . . 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 35157 . . . . . . . . . . . . 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 15808 . . . . . . . . . . . 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 35062 . . . . . . . . . . 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 2457 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
35 eqid 2457 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
362, 34, 35op0le 35054 . . . . . . . . . 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 35138 . . . . . . . . . . 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 35072 . . . . . . . . . . 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 35233 . . . . . . . . . . 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 35052 . . . . . . . . . . 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 15729 . . . . . . . . . 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 15719 . . . . . . . . 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 2728 . . . . . 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 35303 . . . . . 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 35138 . . . . . . . . . . 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 35072 . . . . . . . . . . 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 35063 . . . . . . . . . 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 4480 . . . . . . . 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 35157 . . . . . . . . 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 15729 . . . . . . . 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 2932 . . . . 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 2955 . 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 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   occoc 14720   Posetcpo 15696   ltcplt 15697   joincjn 15700   0.cp0 15794   1.cp1 15795   Latclat 15802   OPcops 35040    <o ccvr 35130   Atomscatm 35131   HLchlt 35218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219
This theorem is referenced by:  1cvratlt  35341  lhpexlt  35869
  Copyright terms: Public domain W3C validator