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Theorem 4atex 30265
Description: Whenever there are at least 4 atoms under  P  .\/  Q (specifically,  P,  Q,  r, and  ( P  .\/  Q
)  ./\  W), there are also at least 4 atoms under  P  .\/  S. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p  \/ q/0 and hence p  \/ s/0 contains at least four atoms..." Note that by cvlsupr2 29533, our  ( P  .\/  r )  =  ( Q  .\/  r ) is a shorter way to express  r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ). (Contributed by NM, 27-May-2013.)
Hypotheses
Ref Expression
4that.l  |-  .<_  =  ( le `  K )
4that.j  |-  .\/  =  ( join `  K )
4that.a  |-  A  =  ( Atoms `  K )
4that.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
4atex  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
Distinct variable groups:    z, r, A    H, r    .\/ , r,
z    K, r, z    .<_ , r, z    P, r, z    Q, r, z    S, r, z    W, r, z
Allowed substitution hint:    H( z)

Proof of Theorem 4atex
StepHypRef Expression
1 simp21l 1072 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  e.  A
)
21ad2antrr 706 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  /\  S  =  P )  ->  P  e.  A )
3 simp21r 1073 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  P  .<_  W )
43ad2antrr 706 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  /\  S  =  P )  ->  -.  P  .<_  W )
5 oveq1 5865 . . . . . 6  |-  ( P  =  S  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
65eqcoms 2286 . . . . 5  |-  ( S  =  P  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
76adantl 452 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  /\  S  =  P )  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
8 breq1 4026 . . . . . . 7  |-  ( z  =  P  ->  (
z  .<_  W  <->  P  .<_  W ) )
98notbid 285 . . . . . 6  |-  ( z  =  P  ->  ( -.  z  .<_  W  <->  -.  P  .<_  W ) )
10 oveq2 5866 . . . . . . 7  |-  ( z  =  P  ->  ( P  .\/  z )  =  ( P  .\/  P
) )
11 oveq2 5866 . . . . . . 7  |-  ( z  =  P  ->  ( S  .\/  z )  =  ( S  .\/  P
) )
1210, 11eqeq12d 2297 . . . . . 6  |-  ( z  =  P  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  P )  =  ( S  .\/  P ) ) )
139, 12anbi12d 691 . . . . 5  |-  ( z  =  P  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  P  .<_  W  /\  ( P 
.\/  P )  =  ( S  .\/  P
) ) ) )
1413rspcev 2884 . . . 4  |-  ( ( P  e.  A  /\  ( -.  P  .<_  W  /\  ( P  .\/  P )  =  ( S 
.\/  P ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
152, 4, 7, 14syl12anc 1180 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  /\  S  =  P )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
16 simpl3r 1011 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
1716ad2antrr 706 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =  Q )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
18 oveq1 5865 . . . . . . . . . 10  |-  ( S  =  Q  ->  ( S  .\/  z )  =  ( Q  .\/  z
) )
1918eqeq2d 2294 . . . . . . . . 9  |-  ( S  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2019anbi2d 684 . . . . . . . 8  |-  ( S  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2120rexbidv 2564 . . . . . . 7  |-  ( S  =  Q  ->  ( E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( Q  .\/  z ) ) ) )
22 breq1 4026 . . . . . . . . . 10  |-  ( r  =  z  ->  (
r  .<_  W  <->  z  .<_  W ) )
2322notbid 285 . . . . . . . . 9  |-  ( r  =  z  ->  ( -.  r  .<_  W  <->  -.  z  .<_  W ) )
24 oveq2 5866 . . . . . . . . . 10  |-  ( r  =  z  ->  ( P  .\/  r )  =  ( P  .\/  z
) )
25 oveq2 5866 . . . . . . . . . 10  |-  ( r  =  z  ->  ( Q  .\/  r )  =  ( Q  .\/  z
) )
2624, 25eqeq12d 2297 . . . . . . . . 9  |-  ( r  =  z  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2723, 26anbi12d 691 . . . . . . . 8  |-  ( r  =  z  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2827cbvrexv 2765 . . . . . . 7  |-  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( Q  .\/  z
) ) )
2921, 28syl6rbbr 255 . . . . . 6  |-  ( S  =  Q  ->  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )
3029adantl 452 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =  Q )  ->  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )
3117, 30mpbid 201 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =  Q )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
32 simp22l 1074 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  Q  e.  A
)
3332ad3antrrr 710 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  Q  e.  A )
34 simp22r 1075 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  Q  .<_  W )
3534ad3antrrr 710 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  -.  Q  .<_  W )
36 simp3l 983 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  =/=  Q
)
3736necomd 2529 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  Q  =/=  P
)
3837ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  Q  =/=  P )
39 simpr 447 . . . . . . 7  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  S  =/=  Q )
4039necomd 2529 . . . . . 6  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  Q  =/=  S )
41 simpllr 735 . . . . . . 7  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  S  .<_  ( P  .\/  Q ) )
42 simp1l 979 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  HL )
43 hlcvl 29549 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CvLat )
4442, 43syl 15 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  CvLat )
4544ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  K  e.  CvLat
)
46 simp23 990 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  S  e.  A
)
4746ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  S  e.  A )
481ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  P  e.  A )
49 simplr 731 . . . . . . . 8  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  S  =/=  P )
50 4that.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
51 4that.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
52 4that.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
5350, 51, 52cvlatexch1 29526 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( S  e.  A  /\  Q  e.  A  /\  P  e.  A )  /\  S  =/=  P
)  ->  ( S  .<_  ( P  .\/  Q
)  ->  Q  .<_  ( P  .\/  S ) ) )
5445, 47, 33, 48, 49, 53syl131anc 1195 . . . . . . 7  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  ( S  .<_  ( P  .\/  Q
)  ->  Q  .<_  ( P  .\/  S ) ) )
5541, 54mpd 14 . . . . . 6  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  Q  .<_  ( P  .\/  S ) )
5649necomd 2529 . . . . . . 7  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  P  =/=  S )
5752, 50, 51cvlsupr2 29533 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  Q  e.  A )  /\  P  =/=  S
)  ->  ( ( P  .\/  Q )  =  ( S  .\/  Q
)  <->  ( Q  =/= 
P  /\  Q  =/=  S  /\  Q  .<_  ( P 
.\/  S ) ) ) )
5845, 48, 47, 33, 56, 57syl131anc 1195 . . . . . 6  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  ( ( P  .\/  Q )  =  ( S  .\/  Q
)  <->  ( Q  =/= 
P  /\  Q  =/=  S  /\  Q  .<_  ( P 
.\/  S ) ) ) )
5938, 40, 55, 58mpbir3and 1135 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  ( P  .\/  Q )  =  ( S  .\/  Q ) )
60 breq1 4026 . . . . . . . 8  |-  ( z  =  Q  ->  (
z  .<_  W  <->  Q  .<_  W ) )
6160notbid 285 . . . . . . 7  |-  ( z  =  Q  ->  ( -.  z  .<_  W  <->  -.  Q  .<_  W ) )
62 oveq2 5866 . . . . . . . 8  |-  ( z  =  Q  ->  ( P  .\/  z )  =  ( P  .\/  Q
) )
63 oveq2 5866 . . . . . . . 8  |-  ( z  =  Q  ->  ( S  .\/  z )  =  ( S  .\/  Q
) )
6462, 63eqeq12d 2297 . . . . . . 7  |-  ( z  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  Q )  =  ( S  .\/  Q ) ) )
6561, 64anbi12d 691 . . . . . 6  |-  ( z  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  Q  .<_  W  /\  ( P 
.\/  Q )  =  ( S  .\/  Q
) ) ) )
6665rspcev 2884 . . . . 5  |-  ( ( Q  e.  A  /\  ( -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( S 
.\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6733, 35, 59, 66syl12anc 1180 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  S  .<_  ( P 
.\/  Q ) )  /\  S  =/=  P
)  /\  S  =/=  Q )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6831, 67pm2.61dane 2524 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  /\  S  =/= 
P )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6915, 68pm2.61dane 2524 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  S  .<_  ( P  .\/  Q ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
70 simpl1 958 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
71 simpl2 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )
72 simpl3l 1010 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  Q )
73 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  -.  S  .<_  ( P  .\/  Q ) )
74 simpl3r 1011 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
75 4that.h . . . 4  |-  H  =  ( LHyp `  K
)
7650, 51, 52, 754atexlem7 30264 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
7770, 71, 72, 73, 74, 76syl113anc 1194 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
7869, 77pm2.61dan 766 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   CvLatclc 29455   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  4atex2  30266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lhyp 30177
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