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Theorem cdleme0moN 29318
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 2497 . . 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 28453 . . . . . . . . 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 28437 . . . . . . . 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 29138 . . . . . . . 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 29315 . . . . . . . 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 5718 . . . . . . . 8  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
40 oveq2 5718 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
4139, 40eqeq12d 2267 . . . . . . 7  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
42 oveq2 5718 . . . . . . . 8  |-  ( r  =  U  ->  ( P  .\/  r )  =  ( P  .\/  U
) )
43 oveq2 5718 . . . . . . . 8  |-  ( r  =  U  ->  ( Q  .\/  r )  =  ( Q  .\/  U
) )
4442, 43eqeq12d 2267 . . . . . . 7  |-  ( r  =  U  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  U )  =  ( Q  .\/  U ) ) )
4541, 44rmoi 3008 . . . . . 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 )
463, 5, 22, 32, 38, 45syl122anc 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 )
4736simprd 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  U  .<_  W )
4833, 34, 35, 16, 47syl121anc 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 )
4946, 48eqbrtrd 3940 . . . 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 )
5049ex 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 ) )
512, 50syl5bir 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 )
)
521, 51mt3d 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 2115    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   CvLatclc 28359   HLchlt 28444   LHypclh 29077
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081
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