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Theorem cdlemb 34590
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 1026 . . 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 1027 . . . 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 1028 . . . 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 1022 . . . 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 1023 . . . 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 1032 . . . 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 1033 . . . 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 2467 . . . . 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 34272 . . . 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 1251 . . 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 34160 . . . . . 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 34086 . . . . . . 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 34086 . . . . . . 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 15531 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2418, 20, 22, 23syl3anc 1228 . . . . 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 15557 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
) ( meet `  K
) X )  .<_  X )
2618, 24, 4, 25syl3anc 1228 . . . 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 2467 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
288, 9, 27, 12, 13, 141cvratlt 34270 . . . 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 1236 . . 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 34235 . . 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 1231 . 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 1071 . . . 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 1072 . . . 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 755 . . . 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 1078 . . . . 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 764 . . . . . . . 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 1049 . . . . . . . . 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 15441 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  u  e.  A  /\  X  e.  B )  ->  ( u ( lt
`  K ) X  ->  u  .<_  X ) )
3932, 34, 37, 38syl3anc 1228 . . . . . . . 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 4450 . . . . . . 7  |-  ( P  =  u  ->  ( P  .<_  X  <->  u  .<_  X ) )
4240, 41syl5ibrcom 222 . . . . . 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 2679 . . . . 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 34182 . . . 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 1231 . . 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 2467 . . . . . . . 8  |-  ( ( P  .\/  Q ) ( meet `  K
) X )  =  ( ( P  .\/  Q ) ( meet `  K
) X )
488, 9, 10, 12, 13, 14, 27, 11, 47cdlemblem 34589 . . . . . . 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 1195 . . . . . 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 606 . . . . 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 429 . . . 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 2935 . . 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 2959 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   ltcplt 15421   joincjn 15424   meetcmee 15425   1.cp1 15518   Latclat 15525    <o ccvr 34059   Atomscatm 34060   HLchlt 34147
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  cdlemb2  34837
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