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Theorem cdleme0moN 31036
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 1077 . 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 2544 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
3 simpl33 1038 . . . . . 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 1076 . . . . . . 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 451 . . . . . 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 732 . . . . . . 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 733 . . . . . . 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 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 ) )  ->  R  .<_  ( P  .\/  Q ) )
9 simpl1l 1006 . . . . . . . . 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 30171 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  CvLat )
119, 10syl 15 . . . . . . . 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 1072 . . . . . . . . 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 451 . . . . . . . 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 1074 . . . . . . . . 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 451 . . . . . . . 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 1036 . . . . . . . 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 30155 . . . . . . . 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 1195 . . . . . . 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 1135 . . . . . 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 979 . . . . . . . 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 980 . . . . . . . 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 1073 . . . . . . . 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 991 . . . . . . . 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 30856 . . . . . . . 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 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
) ) ) )  ->  U  e.  A
)
3231adantr 451 . . . . . 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 958 . . . . . . 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 1033 . . . . . . 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 1034 . . . . . . 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 31033 . . . . . . . 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 445 . . . . . . 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 1187 . . . . . 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 5882 . . . . . . . . 9  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
40 oveq2 5882 . . . . . . . . 9  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
4139, 40eqeq12d 2310 . . . . . . . 8  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
42 oveq2 5882 . . . . . . . . 9  |-  ( r  =  U  ->  ( P  .\/  r )  =  ( P  .\/  U
) )
43 oveq2 5882 . . . . . . . . 9  |-  ( r  =  U  ->  ( Q  .\/  r )  =  ( Q  .\/  U
) )
4442, 43eqeq12d 2310 . . . . . . . 8  |-  ( r  =  U  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  U )  =  ( Q  .\/  U ) ) )
4541, 44rmoi 3093 . . . . . . 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 2564 . . . . . . . . 9  |-  ( E* r  e.  A ( P  .\/  r )  =  ( Q  .\/  r )  <->  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) )
47463anbi1i 1142 . . . . . . . 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 315 . . . . . . 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 199 . . . . . 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 1191 . . . . 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 449 . . . . . 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 1187 . . . . 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 4059 . . . 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 423 . . 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 209 . 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 117 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E*wmo 2157    =/= wne 2459   E*wrmo 2559   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   CvLatclc 30077   HLchlt 30162   LHypclh 30795
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-rmo 2564  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  df-lhyp 30799
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