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Theorem cdlemblem 34598
Description: Lemma for cdlemb 34599. (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 )
cdlemblem.s  |-  .<  =  ( lt `  K )
cdlemblem.m  |-  ./\  =  ( meet `  K )
cdlemblem.v  |-  V  =  ( ( P  .\/  Q )  ./\  X )
Assertion
Ref Expression
cdlemblem  |-  ( ( ( ( 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )

Proof of Theorem cdlemblem
StepHypRef Expression
1 simp132 1132 . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  -.  P  .<_  X )
2 simp111 1125 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  K  e.  HL )
3 simp2l 1022 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  u  e.  A )
4 simp12l 1109 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  X  e.  B )
52, 3, 43jca 1176 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( K  e.  HL  /\  u  e.  A  /\  X  e.  B )
)
6 simp2rr 1066 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  u  .<  X )
7 cdlemb.l . . . . . . 7  |-  .<_  =  ( le `  K )
8 cdlemblem.s . . . . . . 7  |-  .<  =  ( lt `  K )
97, 8pltle 15447 . . . . . 6  |-  ( ( K  e.  HL  /\  u  e.  A  /\  X  e.  B )  ->  ( u  .<  X  ->  u  .<_  X ) )
105, 6, 9sylc 60 . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  u  .<_  X )
11 hllat 34169 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
122, 11syl 16 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  K  e.  Lat )
13 simp3l 1024 . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
r  e.  A )
14 cdlemb.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
15 cdlemb.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1614, 15atbase 34095 . . . . . . . 8  |-  ( r  e.  A  ->  r  e.  B )
1713, 16syl 16 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
r  e.  B )
1814, 15atbase 34095 . . . . . . . 8  |-  ( u  e.  A  ->  u  e.  B )
193, 18syl 16 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  u  e.  B )
20 cdlemb.j . . . . . . . 8  |-  .\/  =  ( join `  K )
2114, 7, 20latjle12 15548 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( r  e.  B  /\  u  e.  B  /\  X  e.  B
) )  ->  (
( r  .<_  X  /\  u  .<_  X )  <->  ( r  .\/  u )  .<_  X ) )
2212, 17, 19, 4, 21syl13anc 1230 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( r  .<_  X  /\  u  .<_  X )  <-> 
( r  .\/  u
)  .<_  X ) )
2322biimpd 207 . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( r  .<_  X  /\  u  .<_  X )  ->  ( r  .\/  u )  .<_  X ) )
2410, 23mpan2d 674 . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  X  -> 
( r  .\/  u
)  .<_  X ) )
25 simp112 1126 . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  P  e.  A )
2613, 25, 33jca 1176 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  e.  A  /\  P  e.  A  /\  u  e.  A
) )
27 simp3r2 1105 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
r  =/=  u )
282, 26, 273jca 1176 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( K  e.  HL  /\  ( r  e.  A  /\  P  e.  A  /\  u  e.  A
)  /\  r  =/=  u ) )
29 simp3r3 1106 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
r  .<_  ( P  .\/  u ) )
307, 20, 15hlatexch2 34201 . . . . . 6  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  P  e.  A  /\  u  e.  A
)  /\  r  =/=  u )  ->  (
r  .<_  ( P  .\/  u )  ->  P  .<_  ( r  .\/  u
) ) )
3128, 29, 30sylc 60 . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  P  .<_  ( r  .\/  u ) )
3214, 15atbase 34095 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
3325, 32syl 16 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  P  e.  B )
3414, 20latjcl 15537 . . . . . . 7  |-  ( ( K  e.  Lat  /\  r  e.  B  /\  u  e.  B )  ->  ( r  .\/  u
)  e.  B )
3512, 17, 19, 34syl3anc 1228 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .\/  u
)  e.  B )
3614, 7lattr 15542 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  ( r  .\/  u
)  e.  B  /\  X  e.  B )
)  ->  ( ( P  .<_  ( r  .\/  u )  /\  (
r  .\/  u )  .<_  X )  ->  P  .<_  X ) )
3712, 33, 35, 4, 36syl13anc 1230 . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( P  .<_  ( r  .\/  u )  /\  ( r  .\/  u )  .<_  X )  ->  P  .<_  X ) )
3831, 37mpand 675 . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( r  .\/  u )  .<_  X  ->  P  .<_  X ) )
3924, 38syld 44 . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  X  ->  P  .<_  X ) )
401, 39mtod 177 . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  -.  r  .<_  X )
41 simp2rl 1065 . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  u  =/=  V )
42 simp113 1127 . . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  Q  e.  A )
43 simp3r1 1104 . . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
r  =/=  P )
447, 20, 15hlatexchb1 34198 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  Q  e.  A  /\  P  e.  A
)  /\  r  =/=  P )  ->  ( r  .<_  ( P  .\/  Q
)  <->  ( P  .\/  r )  =  ( P  .\/  Q ) ) )
452, 13, 42, 25, 43, 44syl131anc 1241 . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  ( P 
.\/  Q )  <->  ( P  .\/  r )  =  ( P  .\/  Q ) ) )
4613, 3, 253jca 1176 . . . . . . . . . . 11  |-  ( ( ( ( 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  e.  A  /\  u  e.  A  /\  P  e.  A
) )
472, 46, 433jca 1176 . . . . . . . . . 10  |-  ( ( ( ( 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( K  e.  HL  /\  ( r  e.  A  /\  u  e.  A  /\  P  e.  A
)  /\  r  =/=  P ) )
487, 20, 15hlatexch1 34200 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  u  e.  A  /\  P  e.  A
)  /\  r  =/=  P )  ->  ( r  .<_  ( P  .\/  u
)  ->  u  .<_  ( P  .\/  r ) ) )
4947, 29, 48sylc 60 . . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  u  .<_  ( P  .\/  r ) )
50 breq2 4451 . . . . . . . . 9  |-  ( ( P  .\/  r )  =  ( P  .\/  Q )  ->  ( u  .<_  ( P  .\/  r
)  <->  u  .<_  ( P 
.\/  Q ) ) )
5149, 50syl5ibcom 220 . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( P  .\/  r )  =  ( P  .\/  Q )  ->  u  .<_  ( P 
.\/  Q ) ) )
5245, 51sylbid 215 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  ( P 
.\/  Q )  ->  u  .<_  ( P  .\/  Q ) ) )
5352, 10jctird 544 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  ( P 
.\/  Q )  -> 
( u  .<_  ( P 
.\/  Q )  /\  u  .<_  X ) ) )
5414, 15atbase 34095 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  B )
5542, 54syl 16 . . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  Q  e.  B )
5614, 20latjcl 15537 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
5712, 33, 55, 56syl3anc 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( P  .\/  Q
)  e.  B )
58 cdlemblem.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
5914, 7, 58latlem12 15564 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( u  e.  B  /\  ( P  .\/  Q
)  e.  B  /\  X  e.  B )
)  ->  ( (
u  .<_  ( P  .\/  Q )  /\  u  .<_  X )  <->  u  .<_  ( ( P  .\/  Q ) 
./\  X ) ) )
6012, 19, 57, 4, 59syl13anc 1230 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( u  .<_  ( P  .\/  Q )  /\  u  .<_  X )  <-> 
u  .<_  ( ( P 
.\/  Q )  ./\  X ) ) )
61 cdlemblem.v . . . . . . . 8  |-  V  =  ( ( P  .\/  Q )  ./\  X )
6261breq2i 4455 . . . . . . 7  |-  ( u 
.<_  V  <->  u  .<_  ( ( P  .\/  Q ) 
./\  X ) )
6360, 62syl6bbr 263 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( u  .<_  ( P  .\/  Q )  /\  u  .<_  X )  <-> 
u  .<_  V ) )
6453, 63sylibd 214 . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  ( P 
.\/  Q )  ->  u  .<_  V ) )
65 hlatl 34166 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
662, 65syl 16 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  K  e.  AtLat )
67 simp12r 1110 . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  P  =/=  Q )
68 simp131 1131 . . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  X C  .1.  )
69 cdlemb.u . . . . . . . . 9  |-  .1.  =  ( 1. `  K )
70 cdlemb.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
7114, 7, 20, 58, 69, 70, 151cvrat 34281 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )
722, 25, 42, 4, 67, 68, 1, 71syl133anc 1251 . . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )
7361, 72syl5eqel 2559 . . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  V  e.  A )
747, 15atcmp 34117 . . . . . 6  |-  ( ( K  e.  AtLat  /\  u  e.  A  /\  V  e.  A )  ->  (
u  .<_  V  <->  u  =  V ) )
7566, 3, 73, 74syl3anc 1228 . . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( u  .<_  V  <->  u  =  V ) )
7664, 75sylibd 214 . . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( r  .<_  ( P 
.\/  Q )  ->  u  =  V )
)
7776necon3ad 2677 . . 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( u  =/=  V  ->  -.  r  .<_  ( P 
.\/  Q ) ) )
7841, 77mpd 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  -.  r  .<_  ( P 
.\/  Q ) )
7940, 78jca 532 1  |-  ( ( ( ( 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   Basecbs 14489   lecple 14561   ltcplt 15427   joincjn 15430   meetcmee 15431   1.cp1 15524   Latclat 15531    <o ccvr 34068   Atomscatm 34069   AtLatcal 34070   HLchlt 34156
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 6575
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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-poset 15432  df-plt 15444  df-lub 15460  df-glb 15461  df-join 15462  df-meet 15463  df-p0 15525  df-p1 15526  df-lat 15532  df-clat 15594  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157
This theorem is referenced by:  cdlemb  34599
  Copyright terms: Public domain W3C validator