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Theorem cdlemb 30322
Description: Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
Hypotheses
Ref Expression
cdlemb.b  |-  B  =  ( Base `  K
)
cdlemb.l  |-  .<_  =  ( le `  K )
cdlemb.j  |-  .\/  =  ( join `  K )
cdlemb.u  |-  .1.  =  ( 1. `  K )
cdlemb.c  |-  C  =  (  <o  `  K )
cdlemb.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlemb  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    B, r    C, r    .\/ , r    K, r    .<_ , r    P, r    Q, r    .1. , r    X, r

Proof of Theorem cdlemb
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  HL )
2 simp12 988 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  A )
3 simp13 989 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  A )
4 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  X  e.  B )
5 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
6 simp31 993 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  X C  .1.  )
7 simp32 994 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  P  .<_  X )
8 cdlemb.b . . . . 5  |-  B  =  ( Base `  K
)
9 cdlemb.l . . . . 5  |-  .<_  =  ( le `  K )
10 cdlemb.j . . . . 5  |-  .\/  =  ( join `  K )
11 eqid 2430 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
12 cdlemb.u . . . . 5  |-  .1.  =  ( 1. `  K )
13 cdlemb.c . . . . 5  |-  C  =  (  <o  `  K )
14 cdlemb.a . . . . 5  |-  A  =  ( Atoms `  K )
158, 9, 10, 11, 12, 13, 141cvrat 30004 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  e.  A )
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1207 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  e.  A )
17 hllat 29892 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
181, 17syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  Lat )
198, 14atbase 29818 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
202, 19syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  B )
218, 14atbase 29818 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
223, 21syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  B )
238, 10latjcl 14462 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2418, 20, 22, 23syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( P  .\/  Q
)  e.  B )
258, 9, 11latmle2 14489 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
) ( meet `  K
) X )  .<_  X )
2618, 24, 4, 25syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  .<_  X )
27 eqid 2430 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
288, 9, 27, 12, 13, 141cvratlt 30002 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  Q ) ( meet `  K
) X )  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  (
( P  .\/  Q
) ( meet `  K
) X )  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X ) ( lt `  K ) X )
291, 16, 4, 6, 26, 28syl32anc 1192 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X ) ( lt `  K ) X )
308, 27, 142atlt 29967 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  Q ) ( meet `  K
) X )  e.  A  /\  X  e.  B )  /\  (
( P  .\/  Q
) ( meet `  K
) X ) ( lt `  K ) X )  ->  E. u  e.  A  ( u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) )
311, 16, 4, 29, 30syl31anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. u  e.  A  ( u  =/=  (
( P  .\/  Q
) ( meet `  K
) X )  /\  u ( lt `  K ) X ) )
32 simpl11 1032 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  K  e.  HL )
33 simpl12 1033 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  P  e.  A )
34 simprl 733 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  u  e.  A )
35 simpl32 1039 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  -.  P  .<_  X )
36 simprrr 742 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  u
( lt `  K
) X )
37 simpl2l 1010 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  X  e.  B )
389, 27pltle 14401 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  u  e.  A  /\  X  e.  B )  ->  ( u ( lt
`  K ) X  ->  u  .<_  X ) )
3932, 34, 37, 38syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  (
u ( lt `  K ) X  ->  u  .<_  X ) )
4036, 39mpd 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  u  .<_  X )
41 breq1 4202 . . . . . . 7  |-  ( P  =  u  ->  ( P  .<_  X  <->  u  .<_  X ) )
4240, 41syl5ibrcom 214 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  ( P  =  u  ->  P 
.<_  X ) )
4342necon3bd 2630 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  ( -.  P  .<_  X  ->  P  =/=  u ) )
4435, 43mpd 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  P  =/=  u )
459, 10, 14hlsupr 29914 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  u  e.  A )  /\  P  =/=  u
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) )
4632, 33, 34, 44, 45syl31anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) )
47 eqid 2430 . . . . . . . 8  |-  ( ( P  .\/  Q ) ( meet `  K
) X )  =  ( ( P  .\/  Q ) ( meet `  K
) X )
488, 9, 10, 12, 13, 14, 27, 11, 47cdlemblem 30321 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) )  /\  ( r  e.  A  /\  (
r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u
) ) ) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q
) ) )
49483exp 1152 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( u  e.  A  /\  ( u  =/=  ( ( P 
.\/  Q ) (
meet `  K ) X )  /\  u
( lt `  K
) X ) )  ->  ( ( r  e.  A  /\  (
r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u
) ) )  -> 
( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) ) ) )
5049exp4a 590 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( u  e.  A  /\  ( u  =/=  ( ( P 
.\/  Q ) (
meet `  K ) X )  /\  u
( lt `  K
) X ) )  ->  ( r  e.  A  ->  ( (
r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u
) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P 
.\/  Q ) ) ) ) ) )
5150imp 419 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  (
r  e.  A  -> 
( ( r  =/= 
P  /\  r  =/=  u  /\  r  .<_  ( P 
.\/  u ) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q
) ) ) ) )
5251reximdvai 2803 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  ( E. r  e.  A  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P 
.\/  u ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
5346, 52mpd 15 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
5431, 53rexlimddv 2821 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   E.wrex 2693   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   ltcplt 14381   joincjn 14384   meetcmee 14385   1.cp1 14450   Latclat 14457    <o ccvr 29791   Atomscatm 29792   HLchlt 29879
This theorem is referenced by:  cdlemb2  30569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880
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