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Theorem cdlemb 35661
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 2457 . . . . 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 35343 . . . 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 35231 . . . . . 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 35157 . . . . . . 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 35157 . . . . . . 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 15808 . . . . . 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 15834 . . . . 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 2457 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
288, 9, 27, 12, 13, 141cvratlt 35341 . . . 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 35306 . . 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 756 . . . 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 766 . . . . . . . 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 15718 . . . . . . . . 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 4459 . . . . . . 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 2669 . . . . 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 35253 . . . 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 2457 . . . . . . . 8  |-  ( ( P  .\/  Q ) ( meet `  K
) X )  =  ( ( P  .\/  Q ) ( meet `  K
) X )
488, 9, 10, 12, 13, 14, 27, 11, 47cdlemblem 35660 . . . . . . 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 2929 . . 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 2953 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 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   ltcplt 15697   joincjn 15700   meetcmee 15701   1.cp1 15795   Latclat 15802    <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:  cdlemb2  35908
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