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Theorem cdleme0moN 31095
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 1080 . 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 2691 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
3 simpl33 1041 . . . . . 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 1079 . . . . . . 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 734 . . . . . . 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 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  =/=  Q )
8 simpl32 1040 . . . . . . 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 1009 . . . . . . . . 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 30230 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  CvLat )
119, 10syl 16 . . . . . . . 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 1075 . . . . . . . . 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 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
) ) ) )  ->  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 1039 . . . . . . . 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 30214 . . . . . . . 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 1198 . . . . . . 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 1138 . . . . . 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 982 . . . . . . . 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 983 . . . . . . . 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 1076 . . . . . . . 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 994 . . . . . . . 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 30915 . . . . . . . 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 1201 . . . . . . 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 961 . . . . . . 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 1036 . . . . . . 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 1037 . . . . . . 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 31092 . . . . . . . 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 1190 . . . . . 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 df-rmo 2715 . . . . . . 7  |-  ( E* r  e.  A ( P  .\/  r )  =  ( Q  .\/  r )  <->  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) )
40 oveq2 6092 . . . . . . . . 9  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
41 oveq2 6092 . . . . . . . . 9  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
4240, 41eqeq12d 2452 . . . . . . . 8  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
43 oveq2 6092 . . . . . . . . 9  |-  ( r  =  U  ->  ( P  .\/  r )  =  ( P  .\/  U
) )
44 oveq2 6092 . . . . . . . . 9  |-  ( r  =  U  ->  ( Q  .\/  r )  =  ( Q  .\/  U
) )
4543, 44eqeq12d 2452 . . . . . . . 8  |-  ( r  =  U  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  U )  =  ( Q  .\/  U ) ) )
4642, 45rmoi 3252 . . . . . . 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 )
4739, 46syl3an1br 1224 . . . . . 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 )
483, 5, 22, 32, 38, 47syl122anc 1194 . . . . 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 )
4936simprd 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  U  .<_  W )
5033, 34, 35, 16, 49syl121anc 1190 . . . . 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 )
5148, 50eqbrtrd 4235 . . . 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 )
5251ex 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 ) )
532, 52syl5bir 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 )
)
541, 53mt3d 120 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 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E*wmo 2284    =/= wne 2601   E*wrmo 2710   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30134   CvLatclc 30136   HLchlt 30221   LHypclh 30854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30047  df-ol 30049  df-oml 30050  df-covers 30137  df-ats 30138  df-atl 30169  df-cvlat 30193  df-hlat 30222  df-lhyp 30858
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