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Theorem cdlemblem 30604
Description: Lemma for cdlemb 30605. (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 1091 . . 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 1084 . . . . . . 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 981 . . . . . . 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 1068 . . . . . . 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 1132 . . . . . 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 1025 . . . . . 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 14111 . . . . . 6  |-  ( ( K  e.  HL  /\  u  e.  A  /\  X  e.  B )  ->  ( u  .<  X  ->  u  .<_  X ) )
105, 6, 9sylc 56 . . . . 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 30175 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
122, 11syl 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  ->  K  e.  Lat )
13 simp3l 983 . . . . . . . 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 30101 . . . . . . . 8  |-  ( r  e.  A  ->  r  e.  B )
1713, 16syl 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  =/=  V  /\  u  .<  X ) )  /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) ) )  -> 
r  e.  B )
1814, 15atbase 30101 . . . . . . . 8  |-  ( u  e.  A  ->  u  e.  B )
193, 18syl 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  =/=  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 14184 . . . . . . 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 1184 . . . . . 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 198 . . . . 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 655 . . . 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 1085 . . . . . . . 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 1132 . . . . . . 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 1064 . . . . . . 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 1132 . . . . . 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 1065 . . . . . 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 30207 . . . . . 6  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  P  e.  A  /\  u  e.  A
)  /\  r  =/=  u )  ->  (
r  .<_  ( P  .\/  u )  ->  P  .<_  ( r  .\/  u
) ) )
3128, 29, 30sylc 56 . . . . 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 30101 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
3325, 32syl 15 . . . . . 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 14172 . . . . . . 7  |-  ( ( K  e.  Lat  /\  r  e.  B  /\  u  e.  B )  ->  ( r  .\/  u
)  e.  B )
3512, 17, 19, 34syl3anc 1182 . . . . . 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 14178 . . . . . 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 1184 . . . . 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 656 . . . 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 40 . . 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 168 . 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 1024 . . 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 1086 . . . . . . . . 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 1063 . . . . . . . . 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 30204 . . . . . . . . 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 1195 . . . . . . . 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 1132 . . . . . . . . . . 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 1132 . . . . . . . . . 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 30206 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  u  e.  A  /\  P  e.  A
)  /\  r  =/=  P )  ->  ( r  .<_  ( P  .\/  u
)  ->  u  .<_  ( P  .\/  r ) ) )
4947, 29, 48sylc 56 . . . . . . . . 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 4043 . . . . . . . . 9  |-  ( ( P  .\/  r )  =  ( P  .\/  Q )  ->  ( u  .<_  ( P  .\/  r
)  <->  u  .<_  ( P 
.\/  Q ) ) )
5149, 50syl5ibcom 211 . . . . . . . 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 206 . . . . . . 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 528 . . . . . 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 30101 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  B )
5542, 54syl 15 . . . . . . . . 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 14172 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
5712, 33, 55, 56syl3anc 1182 . . . . . . . 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 14200 . . . . . . . 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 1184 . . . . . . 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 4047 . . . . . . 7  |-  ( u 
.<_  V  <->  u  .<_  ( ( P  .\/  Q ) 
./\  X ) )
6360, 62syl6bbr 254 . . . . . 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 205 . . . . 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 30172 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
662, 65syl 15 . . . . . 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 1069 . . . . . . . 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 1090 . . . . . . . 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 30287 . . . . . . . 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 1205 . . . . . . 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 2380 . . . . . 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 30123 . . . . . 6  |-  ( ( K  e.  AtLat  /\  u  e.  A  /\  V  e.  A )  ->  (
u  .<_  V  <->  u  =  V ) )
7566, 3, 73, 74syl3anc 1182 . . . . 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 205 . . . 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 2495 . . 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 14 . 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 518 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   ltcplt 14091   joincjn 14094   meetcmee 14095   1.cp1 14160   Latclat 14167    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  cdlemb  30605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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