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Theorem cdleme0moN 29565
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0moN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( R  =  P  \/  R  =  Q ) )
Distinct variable groups:    A, r    .\/ , r    P, r    Q, r    R, r    U, r
Allowed substitution hints:    H( r)    K( r)   
.<_ ( r)    ./\ ( r)    W( r)

Proof of Theorem cdleme0moN
StepHypRef Expression
1 simp23r 1082 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  R  .<_  W )
2 neanior 2504 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
3 simpl33 1043 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) )
4 simp23l 1081 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  R  e.  A
)
54adantr 453 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  R  e.  A )
6 simprl 735 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  R  =/=  P )
7 simprr 736 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  R  =/=  Q )
8 simpl32 1042 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  R  .<_  ( P  .\/  Q ) )
9 simpl1l 1011 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  K  e.  HL )
10 hlcvl 28700 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  CvLat )
119, 10syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  K  e.  CvLat )
12 simp21l 1077 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  e.  A
)
1312adantr 453 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  P  e.  A )
14 simp22l 1079 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  Q  e.  A
)
1514adantr 453 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  Q  e.  A )
16 simpl31 1041 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  P  =/=  Q )
17 cdleme0.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
18 cdleme0.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
19 cdleme0.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
2017, 18, 19cvlsupr2 28684 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
2111, 13, 15, 5, 16, 20syl131anc 1200 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  -> 
( ( P  .\/  R )  =  ( Q 
.\/  R )  <->  ( R  =/=  P  /\  R  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) ) )
226, 7, 8, 21mpbir3and 1140 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) )
23 simp1l 984 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  HL )
24 simp1r 985 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  W  e.  H
)
25 simp21r 1078 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  P  .<_  W )
26 simp31 996 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  =/=  Q
)
27 cdleme0.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
28 cdleme0.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
29 cdleme0.u . . . . . . . . 9  |-  U  =  ( ( P  .\/  Q )  ./\  W )
3018, 19, 27, 17, 28, 29lhpat2 29385 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
3123, 24, 12, 25, 14, 26, 30syl222anc 1203 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  U  e.  A
)
3231adantr 453 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  U  e.  A )
33 simpl1 963 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
34 simpl21 1038 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
35 simpl22 1039 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
3618, 19, 27, 17, 28, 29cdleme02N 29562 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  (
( P  .\/  U
)  =  ( Q 
.\/  U )  /\  U  .<_  W ) )
3736simpld 447 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  ( P  .\/  U )  =  ( Q  .\/  U
) )
3833, 34, 35, 16, 37syl121anc 1192 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  -> 
( P  .\/  U
)  =  ( Q 
.\/  U ) )
39 oveq2 5786 . . . . . . . . 9  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
40 oveq2 5786 . . . . . . . . 9  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
4139, 40eqeq12d 2270 . . . . . . . 8  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
42 oveq2 5786 . . . . . . . . 9  |-  ( r  =  U  ->  ( P  .\/  r )  =  ( P  .\/  U
) )
43 oveq2 5786 . . . . . . . . 9  |-  ( r  =  U  ->  ( Q  .\/  r )  =  ( Q  .\/  U
) )
4442, 43eqeq12d 2270 . . . . . . . 8  |-  ( r  =  U  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  U )  =  ( Q  .\/  U ) ) )
4541, 44rmoi 3041 . . . . . . 7  |-  ( ( E* r  e.  A
( P  .\/  r
)  =  ( Q 
.\/  r )  /\  ( R  e.  A  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) )  /\  ( U  e.  A  /\  ( P 
.\/  U )  =  ( Q  .\/  U
) ) )  ->  R  =  U )
46 df-rmo 2524 . . . . . . . . 9  |-  ( E* r  e.  A ( P  .\/  r )  =  ( Q  .\/  r )  <->  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) )
47463anbi1i 1147 . . . . . . . 8  |-  ( ( E* r  e.  A
( P  .\/  r
)  =  ( Q 
.\/  r )  /\  ( R  e.  A  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) )  /\  ( U  e.  A  /\  ( P 
.\/  U )  =  ( Q  .\/  U
) ) )  <->  ( E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) )  /\  ( R  e.  A  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( U  e.  A  /\  ( P  .\/  U )  =  ( Q  .\/  U ) ) ) )
4847imbi1i 317 . . . . . . 7  |-  ( ( ( E* r  e.  A ( P  .\/  r )  =  ( Q  .\/  r )  /\  ( R  e.  A  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  /\  ( U  e.  A  /\  ( P  .\/  U )  =  ( Q  .\/  U ) ) )  ->  R  =  U )  <->  ( ( E* r ( r  e.  A  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  /\  ( R  e.  A  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) )  /\  ( U  e.  A  /\  ( P 
.\/  U )  =  ( Q  .\/  U
) ) )  ->  R  =  U )
)
4945, 48mpbi 201 . . . . . 6  |-  ( ( E* r ( r  e.  A  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( R  e.  A  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( U  e.  A  /\  ( P  .\/  U )  =  ( Q  .\/  U ) ) )  ->  R  =  U )
503, 5, 22, 32, 38, 49syl122anc 1196 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  R  =  U )
5136simprd 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  U  .<_  W )
5233, 34, 35, 16, 51syl121anc 1192 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  U  .<_  W )
5350, 52eqbrtrd 4003 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) ) )  /\  ( R  =/=  P  /\  R  =/=  Q ) )  ->  R  .<_  W )
5453ex 425 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( R  =/=  P  /\  R  =/=  Q )  ->  R  .<_  W ) )
552, 54syl5bir 211 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( -.  ( R  =  P  \/  R  =  Q )  ->  R  .<_  W )
)
561, 55mt3d 119 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  E* r ( r  e.  A  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( R  =  P  \/  R  =  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E*wmo 2118    =/= wne 2419   E*wrmo 2519   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   CvLatclc 28606   HLchlt 28691   LHypclh 29324
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-lhyp 29328
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