Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atex Unicode version

Theorem 4atex 28954
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 28222, 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 1077 . . . . 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 709 . . . 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 1078 . . . . 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 709 . . . 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 5717 . . . . . 6  |-  ( P  =  S  ->  ( P  .\/  P )  =  ( S  .\/  P
) )
65eqcoms 2256 . . . . 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 3923 . . . . . . 7  |-  ( z  =  P  ->  (
z  .<_  W  <->  P  .<_  W ) )
98notbid 287 . . . . . 6  |-  ( z  =  P  ->  ( -.  z  .<_  W  <->  -.  P  .<_  W ) )
10 oveq2 5718 . . . . . . 7  |-  ( z  =  P  ->  ( P  .\/  z )  =  ( P  .\/  P
) )
11 oveq2 5718 . . . . . . 7  |-  ( z  =  P  ->  ( S  .\/  z )  =  ( S  .\/  P
) )
1210, 11eqeq12d 2267 . . . . . 6  |-  ( z  =  P  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  P )  =  ( S  .\/  P ) ) )
139, 12anbi12d 694 . . . . 5  |-  ( z  =  P  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  P  .<_  W  /\  ( P 
.\/  P )  =  ( S  .\/  P
) ) ) )
1413rcla4ev 2821 . . . 4  |-  ( ( P  e.  A  /\  ( -.  P  .<_  W  /\  ( P  .\/  P )  =  ( S 
.\/  P ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
152, 4, 7, 14syl12anc 1185 . . 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 1016 . . . . . 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 709 . . . . 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 5717 . . . . . . . . . 10  |-  ( S  =  Q  ->  ( S  .\/  z )  =  ( Q  .\/  z
) )
1918eqeq2d 2264 . . . . . . . . 9  |-  ( S  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2019anbi2d 687 . . . . . . . 8  |-  ( S  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2120rexbidv 2528 . . . . . . 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 3923 . . . . . . . . . 10  |-  ( r  =  z  ->  (
r  .<_  W  <->  z  .<_  W ) )
2322notbid 287 . . . . . . . . 9  |-  ( r  =  z  ->  ( -.  r  .<_  W  <->  -.  z  .<_  W ) )
24 oveq2 5718 . . . . . . . . . 10  |-  ( r  =  z  ->  ( P  .\/  r )  =  ( P  .\/  z
) )
25 oveq2 5718 . . . . . . . . . 10  |-  ( r  =  z  ->  ( Q  .\/  r )  =  ( Q  .\/  z
) )
2624, 25eqeq12d 2267 . . . . . . . . 9  |-  ( r  =  z  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  z )  =  ( Q  .\/  z ) ) )
2723, 26anbi12d 694 . . . . . . . 8  |-  ( r  =  z  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  z  .<_  W  /\  ( P 
.\/  z )  =  ( Q  .\/  z
) ) ) )
2827cbvrexv 2709 . . . . . . 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 1079 . . . . . 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 713 . . . . 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 1080 . . . . . 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 713 . . . . 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 988 . . . . . . . 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 2495 . . . . . . 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 713 . . . . . 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 2495 . . . . . 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 738 . . . . . . 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 984 . . . . . . . . . 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 28238 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CvLat )
4442, 43syl 17 . . . . . . . . 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 713 . . . . . . . 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 995 . . . . . . . . 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 713 . . . . . . . 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 713 . . . . . . . 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 734 . . . . . . . 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 28215 . . . . . . . 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 1200 . . . . . . 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 16 . . . . . 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 2495 . . . . . . 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 28222 . . . . . . 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 1200 . . . . . 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 1140 . . . . 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 3923 . . . . . . . 8  |-  ( z  =  Q  ->  (
z  .<_  W  <->  Q  .<_  W ) )
6160notbid 287 . . . . . . 7  |-  ( z  =  Q  ->  ( -.  z  .<_  W  <->  -.  Q  .<_  W ) )
62 oveq2 5718 . . . . . . . 8  |-  ( z  =  Q  ->  ( P  .\/  z )  =  ( P  .\/  Q
) )
63 oveq2 5718 . . . . . . . 8  |-  ( z  =  Q  ->  ( S  .\/  z )  =  ( S  .\/  Q
) )
6462, 63eqeq12d 2267 . . . . . . 7  |-  ( z  =  Q  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  Q )  =  ( S  .\/  Q ) ) )
6561, 64anbi12d 694 . . . . . 6  |-  ( z  =  Q  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  Q  .<_  W  /\  ( P 
.\/  Q )  =  ( S  .\/  Q
) ) ) )
6665rcla4ev 2821 . . . . 5  |-  ( ( Q  e.  A  /\  ( -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( S 
.\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6733, 35, 59, 66syl12anc 1185 . . . 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 2490 . . 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 2490 . 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 963 . . 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 964 . . 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 1015 . . 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 1016 . . 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 28953 . . 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 1199 . 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 769 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   Atomscatm 28142   CvLatclc 28144   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  4atex2  28955
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lhyp 28866
  Copyright terms: Public domain W3C validator