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Theorem cvlsupr2 29458
Description: Two equivalent ways of expressing that  R is a superposition of  P and  Q. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlsupr2.a  |-  A  =  ( Atoms `  K )
cvlsupr2.l  |-  .<_  =  ( le `  K )
cvlsupr2.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
cvlsupr2  |-  ( ( 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 ) ) ) )

Proof of Theorem cvlsupr2
StepHypRef Expression
1 simpl3 962 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  =/=  Q )
21necomd 2633 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  =/=  P )
3 simplr 732 . . . . . . . . 9  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  P )  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
4 oveq2 6028 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( P  .\/  R )  =  ( P  .\/  P
) )
5 oveq2 6028 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( Q  .\/  R )  =  ( Q  .\/  P
) )
64, 5eqeq12d 2401 . . . . . . . . . . 11  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  P )  =  ( Q  .\/  P ) ) )
7 eqcom 2389 . . . . . . . . . . 11  |-  ( ( P  .\/  P )  =  ( Q  .\/  P )  <->  ( Q  .\/  P )  =  ( P 
.\/  P ) )
86, 7syl6bb 253 . . . . . . . . . 10  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( Q  .\/  P )  =  ( P  .\/  P ) ) )
98adantl 453 . . . . . . . . 9  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  P )  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( Q  .\/  P )  =  ( P  .\/  P ) ) )
103, 9mpbid 202 . . . . . . . 8  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  P )  ->  ( Q  .\/  P )  =  ( P  .\/  P
) )
11 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  CvLat )
12 cvllat 29441 . . . . . . . . . . 11  |-  ( K  e.  CvLat  ->  K  e.  Lat )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  Lat )
14 simpl21 1035 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  e.  A )
15 eqid 2387 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
16 cvlsupr2.a . . . . . . . . . . . 12  |-  A  =  ( Atoms `  K )
1715, 16atbase 29404 . . . . . . . . . . 11  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1814, 17syl 16 . . . . . . . . . 10  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  e.  ( Base `  K
) )
19 cvlsupr2.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
2015, 19latjidm 14430 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P  .\/  P
)  =  P )
2113, 18, 20syl2anc 643 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .\/  P )  =  P )
2221adantr 452 . . . . . . . 8  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  P )  ->  ( P  .\/  P )  =  P )
2310, 22eqtrd 2419 . . . . . . 7  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  P )  ->  ( Q  .\/  P )  =  P )
2423ex 424 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  P  ->  ( Q  .\/  P )  =  P ) )
25 simpl22 1036 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  e.  A )
2615, 16atbase 29404 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2725, 26syl 16 . . . . . . . 8  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  e.  ( Base `  K
) )
28 cvlsupr2.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
2915, 28, 19latleeqj1 14419 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( Q  .<_  P  <->  ( Q  .\/  P )  =  P ) )
3013, 27, 18, 29syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .<_  P  <->  ( Q  .\/  P )  =  P ) )
31 cvlatl 29440 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
3211, 31syl 16 . . . . . . . 8  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  AtLat )
3328, 16atcmp 29426 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  P  e.  A )  ->  ( Q  .<_  P  <->  Q  =  P ) )
3432, 25, 14, 33syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .<_  P  <->  Q  =  P ) )
3530, 34bitr3d 247 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  (
( Q  .\/  P
)  =  P  <->  Q  =  P ) )
3624, 35sylibd 206 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  P  ->  Q  =  P ) )
3736necon3d 2588 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  =/=  P  ->  R  =/=  P ) )
382, 37mpd 15 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  =/=  P )
39 simplr 732 . . . . . . . . 9  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  Q )  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
40 oveq2 6028 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( P  .\/  R )  =  ( P  .\/  Q
) )
41 oveq2 6028 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( Q  .\/  R )  =  ( Q  .\/  Q
) )
4240, 41eqeq12d 2401 . . . . . . . . . 10  |-  ( R  =  Q  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  Q )  =  ( Q  .\/  Q ) ) )
4342adantl 453 . . . . . . . . 9  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  Q )  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  Q )  =  ( Q  .\/  Q ) ) )
4439, 43mpbid 202 . . . . . . . 8  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  Q )  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
4515, 19latjidm 14430 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  Q
)  =  Q )
4613, 27, 45syl2anc 643 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .\/  Q )  =  Q )
4746adantr 452 . . . . . . . 8  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  Q )  ->  ( Q  .\/  Q )  =  Q )
4844, 47eqtrd 2419 . . . . . . 7  |-  ( ( ( ( K  e. 
CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  R )  =  ( Q  .\/  R
) )  /\  R  =  Q )  ->  ( P  .\/  Q )  =  Q )
4948ex 424 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  Q  ->  ( P  .\/  Q )  =  Q ) )
5015, 28, 19latleeqj1 14419 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .<_  Q  <->  ( P  .\/  Q )  =  Q ) )
5113, 18, 27, 50syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .<_  Q  <->  ( P  .\/  Q )  =  Q ) )
5228, 16atcmp 29426 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )
5332, 14, 25, 52syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .<_  Q  <->  P  =  Q ) )
5451, 53bitr3d 247 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  (
( P  .\/  Q
)  =  Q  <->  P  =  Q ) )
5549, 54sylibd 206 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  Q  ->  P  =  Q ) )
5655necon3d 2588 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  =/=  Q  ->  R  =/=  Q ) )
571, 56mpd 15 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  =/=  Q )
58 simpl23 1037 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  e.  A )
5915, 16atbase 29404 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
6058, 59syl 16 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  e.  ( Base `  K
) )
6115, 28, 19latlej1 14416 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  R
) )
6213, 27, 60, 61syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  .<_  ( Q  .\/  R
) )
63 simpr 448 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
6462, 63breqtrrd 4179 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  .<_  ( P  .\/  R
) )
6528, 19, 16cvlatexch1 29451 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  Q  =/=  P
)  ->  ( Q  .<_  ( P  .\/  R
)  ->  R  .<_  ( P  .\/  Q ) ) )
6611, 25, 58, 14, 2, 65syl131anc 1197 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .<_  ( P  .\/  R )  ->  R  .<_  ( P  .\/  Q ) ) )
6764, 66mpd 15 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  .<_  ( P  .\/  Q
) )
6838, 57, 673jca 1134 . 2  |-  ( ( ( 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 ) ) )
69 simpr3 965 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q
) )
70 simpl1 960 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  CvLat )
7170, 12syl 16 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
72 simpl21 1035 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  P  e.  A )
7372, 17syl 16 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  P  e.  ( Base `  K
) )
74 simpl22 1036 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  A )
7574, 26syl 16 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  ( Base `  K
) )
7615, 19latjcom 14415 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
7771, 73, 75, 76syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
7877breq2d 4165 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  ( P  .\/  Q )  <->  R  .<_  ( Q 
.\/  P ) ) )
79 simpl23 1037 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
80 simpr2 964 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  =/=  Q )
8128, 19, 16cvlatexch1 29451 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( R  e.  A  /\  P  e.  A  /\  Q  e.  A )  /\  R  =/=  Q
)  ->  ( R  .<_  ( Q  .\/  P
)  ->  P  .<_  ( Q  .\/  R ) ) )
8270, 79, 72, 74, 80, 81syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  ( Q  .\/  P )  ->  P  .<_  ( Q  .\/  R ) ) )
83 simpr1 963 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  =/=  P )
8483necomd 2633 . . . . . 6  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  R )
8528, 19, 16cvlatexchb2 29450 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
8670, 72, 74, 79, 84, 85syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
8782, 86sylibd 206 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  ( Q  .\/  P )  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
8878, 87sylbid 207 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  ( P  .\/  Q )  ->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
8969, 88mpd 15 . 2  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
9068, 89impbida 806 1  |-  ( ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Latclat 14401   Atomscatm 29378   AtLatcal 29379   CvLatclc 29380
This theorem is referenced by:  cvlsupr3  29459  cvlsupr4  29460  cvlsupr5  29461  cvlsupr6  29462  4atexlemex2  30185  4atex  30190  4atex3  30195  cdleme02N  30336  cdleme0ex2N  30338  cdleme0moN  30339  cdleme0nex  30404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-join 14360  df-lat 14402  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437
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