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Theorem cdleme0moN 33866
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 1110 . 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 2695 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
3 simpl33 1071 . . . . . 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 1109 . . . . . . 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 465 . . . . . 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 755 . . . . . . 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 756 . . . . . . 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 1070 . . . . . . 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 1039 . . . . . . . . 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 33001 . . . . . . . . 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 1105 . . . . . . . . 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 465 . . . . . . . 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 1107 . . . . . . . . 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 465 . . . . . . . 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 1069 . . . . . . . 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 32985 . . . . . . . 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 1231 . . . . . . 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 1171 . . . . . 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 1012 . . . . . . . 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 1013 . . . . . . . 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 1106 . . . . . . . 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 1024 . . . . . . . 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 33686 . . . . . . . 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 1234 . . . . . . 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 465 . . . . . 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 991 . . . . . . 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 1066 . . . . . . 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 1067 . . . . . . 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 33863 . . . . . . . 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 459 . . . . . . 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 1223 . . . . . 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 2721 . . . . . . 7  |-  ( E* r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r )  <->  E* r
( r  e.  A  /\  ( P  .\/  r
)  =  ( Q 
.\/  r ) ) )
40 oveq2 6097 . . . . . . . . 9  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
41 oveq2 6097 . . . . . . . . 9  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
4240, 41eqeq12d 2455 . . . . . . . 8  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
43 oveq2 6097 . . . . . . . . 9  |-  ( r  =  U  ->  ( P  .\/  r )  =  ( P  .\/  U
) )
44 oveq2 6097 . . . . . . . . 9  |-  ( r  =  U  ->  ( Q  .\/  r )  =  ( Q  .\/  U
) )
4543, 44eqeq12d 2455 . . . . . . . 8  |-  ( r  =  U  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  U )  =  ( Q  .\/  U ) ) )
4642, 45rmoi 3285 . . . . . . 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 1257 . . . . . 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 1227 . . . . 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 463 . . . . . 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 1223 . . . . 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 4310 . . . 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 434 . . 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 218 . 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 125 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E*wmo 2254    =/= wne 2604   E*wrmo 2716   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   lecple 14243   joincjn 15112   meetcmee 15113   Atomscatm 32905   CvLatclc 32907   HLchlt 32992   LHypclh 33625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-lhyp 33629
This theorem is referenced by: (None)
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