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Theorem 4atex 33717
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 32985, 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 1105 . . . . 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 725 . . . 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 1106 . . . . 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 725 . . . 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 6096 . . . . . 6  |-  ( P  =  S  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
65eqcoms 2444 . . . . 5  |-  ( S  =  P  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
76adantl 466 . . . 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 4293 . . . . . . 7  |-  ( z  =  P  ->  (
z  .<_  W  <->  P  .<_  W ) )
98notbid 294 . . . . . 6  |-  ( z  =  P  ->  ( -.  z  .<_  W  <->  -.  P  .<_  W ) )
10 oveq2 6097 . . . . . . 7  |-  ( z  =  P  ->  ( P  .\/  z )  =  ( P  .\/  P
) )
11 oveq2 6097 . . . . . . 7  |-  ( z  =  P  ->  ( S  .\/  z )  =  ( S  .\/  P
) )
1210, 11eqeq12d 2455 . . . . . 6  |-  ( z  =  P  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  P )  =  ( S  .\/  P ) ) )
139, 12anbi12d 710 . . . . 5  |-  ( z  =  P  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  P  .<_  W  /\  ( P 
.\/  P )  =  ( S  .\/  P
) ) ) )
1413rspcev 3071 . . . 4  |-  ( ( P  e.  A  /\  ( -.  P  .<_  W  /\  ( P  .\/  P )  =  ( S 
.\/  P ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
152, 4, 7, 14syl12anc 1216 . . 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 1044 . . . . . 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 725 . . . . 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 6096 . . . . . . . . . 10  |-  ( S  =  Q  ->  ( S  .\/  z )  =  ( Q  .\/  z
) )
1918eqeq2d 2452 . . . . . . . . 9  |-  ( S  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2019anbi2d 703 . . . . . . . 8  |-  ( S  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2120rexbidv 2734 . . . . . . 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 4293 . . . . . . . . . 10  |-  ( r  =  z  ->  (
r  .<_  W  <->  z  .<_  W ) )
2322notbid 294 . . . . . . . . 9  |-  ( r  =  z  ->  ( -.  r  .<_  W  <->  -.  z  .<_  W ) )
24 oveq2 6097 . . . . . . . . . 10  |-  ( r  =  z  ->  ( P  .\/  r )  =  ( P  .\/  z
) )
25 oveq2 6097 . . . . . . . . . 10  |-  ( r  =  z  ->  ( Q  .\/  r )  =  ( Q  .\/  z
) )
2624, 25eqeq12d 2455 . . . . . . . . 9  |-  ( r  =  z  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2723, 26anbi12d 710 . . . . . . . 8  |-  ( r  =  z  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2827cbvrexv 2946 . . . . . . 7  |-  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( Q  .\/  z
) ) )
2921, 28syl6rbbr 264 . . . . . 6  |-  ( S  =  Q  ->  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )
3029adantl 466 . . . . 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 210 . . . 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 1107 . . . . . 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 729 . . . . 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 1108 . . . . . 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 729 . . . . 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 1016 . . . . . . . 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 2693 . . . . . . 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 729 . . . . . 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 461 . . . . . . 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 2693 . . . . . 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 758 . . . . . . 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 1012 . . . . . . . . . 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 33001 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CvLat )
4442, 43syl 16 . . . . . . . . 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 729 . . . . . . . 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 1023 . . . . . . . . 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 729 . . . . . . . 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 729 . . . . . . . 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 754 . . . . . . . 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 32978 . . . . . . . 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 1231 . . . . . . 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 15 . . . . . 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 2693 . . . . . . 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 32985 . . . . . . 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 1231 . . . . . 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 1171 . . . . 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 4293 . . . . . . . 8  |-  ( z  =  Q  ->  (
z  .<_  W  <->  Q  .<_  W ) )
6160notbid 294 . . . . . . 7  |-  ( z  =  Q  ->  ( -.  z  .<_  W  <->  -.  Q  .<_  W ) )
62 oveq2 6097 . . . . . . . 8  |-  ( z  =  Q  ->  ( P  .\/  z )  =  ( P  .\/  Q
) )
63 oveq2 6097 . . . . . . . 8  |-  ( z  =  Q  ->  ( S  .\/  z )  =  ( S  .\/  Q
) )
6462, 63eqeq12d 2455 . . . . . . 7  |-  ( z  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  Q )  =  ( S  .\/  Q ) ) )
6561, 64anbi12d 710 . . . . . 6  |-  ( z  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  Q  .<_  W  /\  ( P 
.\/  Q )  =  ( S  .\/  Q
) ) ) )
6665rspcev 3071 . . . . 5  |-  ( ( Q  e.  A  /\  ( -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( S 
.\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6733, 35, 59, 66syl12anc 1216 . . . 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 2687 . . 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 2687 . 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 991 . . 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 992 . . 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 1043 . . 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 461 . . 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 1044 . . 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 33716 . . 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 1230 . 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 789 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   E.wrex 2714   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   lecple 14243   joincjn 15112   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-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-llines 33139  df-lplanes 33140  df-lhyp 33629
This theorem is referenced by:  4atex2  33718
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