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Theorem 4atex 30935
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 30203, 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 1075 . . . . 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 708 . . . 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 1076 . . . . 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 708 . . . 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 6090 . . . . . 6  |-  ( P  =  S  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
65eqcoms 2441 . . . . 5  |-  ( S  =  P  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
76adantl 454 . . . 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 4217 . . . . . . 7  |-  ( z  =  P  ->  (
z  .<_  W  <->  P  .<_  W ) )
98notbid 287 . . . . . 6  |-  ( z  =  P  ->  ( -.  z  .<_  W  <->  -.  P  .<_  W ) )
10 oveq2 6091 . . . . . . 7  |-  ( z  =  P  ->  ( P  .\/  z )  =  ( P  .\/  P
) )
11 oveq2 6091 . . . . . . 7  |-  ( z  =  P  ->  ( S  .\/  z )  =  ( S  .\/  P
) )
1210, 11eqeq12d 2452 . . . . . 6  |-  ( z  =  P  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  P )  =  ( S  .\/  P ) ) )
139, 12anbi12d 693 . . . . 5  |-  ( z  =  P  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  P  .<_  W  /\  ( P 
.\/  P )  =  ( S  .\/  P
) ) ) )
1413rspcev 3054 . . . 4  |-  ( ( P  e.  A  /\  ( -.  P  .<_  W  /\  ( P  .\/  P )  =  ( S 
.\/  P ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
152, 4, 7, 14syl12anc 1183 . . 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 1014 . . . . . 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 708 . . . . 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 6090 . . . . . . . . . 10  |-  ( S  =  Q  ->  ( S  .\/  z )  =  ( Q  .\/  z
) )
1918eqeq2d 2449 . . . . . . . . 9  |-  ( S  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2019anbi2d 686 . . . . . . . 8  |-  ( S  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2120rexbidv 2728 . . . . . . 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 4217 . . . . . . . . . 10  |-  ( r  =  z  ->  (
r  .<_  W  <->  z  .<_  W ) )
2322notbid 287 . . . . . . . . 9  |-  ( r  =  z  ->  ( -.  r  .<_  W  <->  -.  z  .<_  W ) )
24 oveq2 6091 . . . . . . . . . 10  |-  ( r  =  z  ->  ( P  .\/  r )  =  ( P  .\/  z
) )
25 oveq2 6091 . . . . . . . . . 10  |-  ( r  =  z  ->  ( Q  .\/  r )  =  ( Q  .\/  z
) )
2624, 25eqeq12d 2452 . . . . . . . . 9  |-  ( r  =  z  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2723, 26anbi12d 693 . . . . . . . 8  |-  ( r  =  z  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2827cbvrexv 2935 . . . . . . 7  |-  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( Q  .\/  z
) ) )
2921, 28syl6rbbr 257 . . . . . 6  |-  ( S  =  Q  ->  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )
3029adantl 454 . . . . 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 203 . . . 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 1077 . . . . . 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 712 . . . . 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 1078 . . . . . 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 712 . . . . 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 986 . . . . . . . 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 2689 . . . . . . 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 712 . . . . . 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 449 . . . . . . 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 2689 . . . . . 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 737 . . . . . . 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 982 . . . . . . . . . 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 30219 . . . . . . . . . 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 712 . . . . . . . 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 993 . . . . . . . . 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 712 . . . . . . . 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 712 . . . . . . . 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 733 . . . . . . . 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 30196 . . . . . . . 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 1198 . . . . . . 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 2689 . . . . . . 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 30203 . . . . . . 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 1198 . . . . . 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 1138 . . . . 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 4217 . . . . . . . 8  |-  ( z  =  Q  ->  (
z  .<_  W  <->  Q  .<_  W ) )
6160notbid 287 . . . . . . 7  |-  ( z  =  Q  ->  ( -.  z  .<_  W  <->  -.  Q  .<_  W ) )
62 oveq2 6091 . . . . . . . 8  |-  ( z  =  Q  ->  ( P  .\/  z )  =  ( P  .\/  Q
) )
63 oveq2 6091 . . . . . . . 8  |-  ( z  =  Q  ->  ( S  .\/  z )  =  ( S  .\/  Q
) )
6462, 63eqeq12d 2452 . . . . . . 7  |-  ( z  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  Q )  =  ( S  .\/  Q ) ) )
6561, 64anbi12d 693 . . . . . 6  |-  ( z  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  Q  .<_  W  /\  ( P 
.\/  Q )  =  ( S  .\/  Q
) ) ) )
6665rspcev 3054 . . . . 5  |-  ( ( Q  e.  A  /\  ( -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( S 
.\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6733, 35, 59, 66syl12anc 1183 . . . 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 2684 . . 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 2684 . 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 961 . . 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 962 . . 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 1013 . . 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 449 . . 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 1014 . . 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 30934 . . 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 1197 . 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 768 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30123   CvLatclc 30125   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  4atex2  30936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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-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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lhyp 30847
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