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Theorem cdlemblem 33327
Description: Lemma for cdlemb 33328. (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 1141 . . 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 1134 . . . . . . 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 1031 . . . . . . 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 1118 . . . . . . 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 1185 . . . . . 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 1075 . . . . . 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 16206 . . . . . 6  |-  ( ( K  e.  HL  /\  u  e.  A  /\  X  e.  B )  ->  ( u  .<  X  ->  u  .<_  X ) )
105, 6, 9sylc 62 . . . . 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 32898 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
122, 11syl 17 . . . . . . 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 1033 . . . . . . . 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 32824 . . . . . . . 8  |-  ( r  e.  A  ->  r  e.  B )
1713, 16syl 17 . . . . . . 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 32824 . . . . . . . 8  |-  ( u  e.  A  ->  u  e.  B )
193, 18syl 17 . . . . . . 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 16307 . . . . . . 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 1266 . . . . . 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 210 . . . . 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 678 . . . 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 1135 . . . . . . . 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 1185 . . . . . . 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 1114 . . . . . . 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 1185 . . . . . 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 1115 . . . . . 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 32930 . . . . . 6  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  P  e.  A  /\  u  e.  A
)  /\  r  =/=  u )  ->  (
r  .<_  ( P  .\/  u )  ->  P  .<_  ( r  .\/  u
) ) )
3128, 29, 30sylc 62 . . . . 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 32824 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
3325, 32syl 17 . . . . . 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 16296 . . . . . . 7  |-  ( ( K  e.  Lat  /\  r  e.  B  /\  u  e.  B )  ->  ( r  .\/  u
)  e.  B )
3512, 17, 19, 34syl3anc 1264 . . . . . 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 16301 . . . . . 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 1266 . . . . 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 679 . . . 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 45 . . 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 180 . 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 1074 . . 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 1136 . . . . . . . . 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 1113 . . . . . . . . 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 32927 . . . . . . . . 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 1277 . . . . . . . 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 1185 . . . . . . . . . . 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 1185 . . . . . . . . . 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 32929 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  u  e.  A  /\  P  e.  A
)  /\  r  =/=  P )  ->  ( r  .<_  ( P  .\/  u
)  ->  u  .<_  ( P  .\/  r ) ) )
4947, 29, 48sylc 62 . . . . . . . . 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 4427 . . . . . . . . 9  |-  ( ( P  .\/  r )  =  ( P  .\/  Q )  ->  ( u  .<_  ( P  .\/  r
)  <->  u  .<_  ( P 
.\/  Q ) ) )
5149, 50syl5ibcom 223 . . . . . . . 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 218 . . . . . . 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 546 . . . . . 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 32824 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  B )
5542, 54syl 17 . . . . . . . . 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 16296 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
5712, 33, 55, 56syl3anc 1264 . . . . . . . 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 16323 . . . . . . . 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 1266 . . . . . . 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 4431 . . . . . . 7  |-  ( u 
.<_  V  <->  u  .<_  ( ( P  .\/  Q ) 
./\  X ) )
6360, 62syl6bbr 266 . . . . . 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 217 . . . . 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 32895 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
662, 65syl 17 . . . . . 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 1119 . . . . . . . 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 1140 . . . . . . . 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 33010 . . . . . . . 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 1287 . . . . . . 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 2511 . . . . . 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 32846 . . . . . 6  |-  ( ( K  e.  AtLat  /\  u  e.  A  /\  V  e.  A )  ->  (
u  .<_  V  <->  u  =  V ) )
7566, 3, 73, 74syl3anc 1264 . . . . 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 217 . . . 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 2630 . . 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 534 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   Basecbs 15120   lecple 15196   ltcplt 16185   joincjn 16188   meetcmee 16189   1.cp1 16283   Latclat 16290    <o ccvr 32797   Atomscatm 32798   AtLatcal 32799   HLchlt 32885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886
This theorem is referenced by:  cdlemb  33328
  Copyright terms: Public domain W3C validator