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Theorem cvlsupr2 32325
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 1000 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  =/=  Q )
21necomd 2672 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  =/=  P )
3 simplr 754 . . . . . . . . 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 6240 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( P  .\/  R )  =  ( P  .\/  P
) )
5 oveq2 6240 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( Q  .\/  R )  =  ( Q  .\/  P
) )
64, 5eqeq12d 2422 . . . . . . . . . . 11  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  P )  =  ( Q  .\/  P ) ) )
7 eqcom 2409 . . . . . . . . . . 11  |-  ( ( P  .\/  P )  =  ( Q  .\/  P )  <->  ( Q  .\/  P )  =  ( P 
.\/  P ) )
86, 7syl6bb 261 . . . . . . . . . 10  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( Q  .\/  P )  =  ( P  .\/  P ) ) )
98adantl 464 . . . . . . . . 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 210 . . . . . . . 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 998 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  CvLat )
12 cvllat 32308 . . . . . . . . . . 11  |-  ( K  e.  CvLat  ->  K  e.  Lat )
1311, 12syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  Lat )
14 simpl21 1073 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  e.  A )
15 eqid 2400 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
16 cvlsupr2.a . . . . . . . . . . . 12  |-  A  =  ( Atoms `  K )
1715, 16atbase 32271 . . . . . . . . . . 11  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1814, 17syl 17 . . . . . . . . . 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 15918 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P  .\/  P
)  =  P )
2113, 18, 20syl2anc 659 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .\/  P )  =  P )
2221adantr 463 . . . . . . . 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 2441 . . . . . . 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 432 . . . . . 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 1074 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  e.  A )
2615, 16atbase 32271 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2725, 26syl 17 . . . . . . . 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 15907 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( Q  .<_  P  <->  ( Q  .\/  P )  =  P ) )
3013, 27, 18, 29syl3anc 1228 . . . . . . 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 32307 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
3211, 31syl 17 . . . . . . . 8  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  AtLat )
3328, 16atcmp 32293 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  P  e.  A )  ->  ( Q  .<_  P  <->  Q  =  P ) )
3432, 25, 14, 33syl3anc 1228 . . . . . . 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 255 . . . . . 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 214 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  P  ->  Q  =  P ) )
3736necon3d 2625 . . . 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 754 . . . . . . . . 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 6240 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( P  .\/  R )  =  ( P  .\/  Q
) )
41 oveq2 6240 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( Q  .\/  R )  =  ( Q  .\/  Q
) )
4240, 41eqeq12d 2422 . . . . . . . . . 10  |-  ( R  =  Q  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  Q )  =  ( Q  .\/  Q ) ) )
4342adantl 464 . . . . . . . . 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 210 . . . . . . . 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 15918 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  Q
)  =  Q )
4613, 27, 45syl2anc 659 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .\/  Q )  =  Q )
4746adantr 463 . . . . . . . 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 2441 . . . . . . 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 432 . . . . . 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 15907 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .<_  Q  <->  ( P  .\/  Q )  =  Q ) )
5113, 18, 27, 50syl3anc 1228 . . . . . . 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 32293 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )
5332, 14, 25, 52syl3anc 1228 . . . . . . 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 255 . . . . . 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 214 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  Q  ->  P  =  Q ) )
5655necon3d 2625 . . . 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 1075 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  e.  A )
5915, 16atbase 32271 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
6058, 59syl 17 . . . . . 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 15904 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  R
) )
6213, 27, 60, 61syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  .<_  ( Q  .\/  R
) )
63 simpr 459 . . . . 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 4418 . . . 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 32318 . . . . 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 1241 . . . 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 1175 . 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 1003 . . 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 998 . . . . . . 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 17 . . . . . 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 1073 . . . . . . 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 17 . . . . . 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 1074 . . . . . . 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 17 . . . . . 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 15903 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
7771, 73, 75, 76syl3anc 1228 . . . . 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 4404 . . . 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 1075 . . . . . 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 1002 . . . . . 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 32318 . . . . . 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 1241 . . . . 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 1001 . . . . . . 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 2672 . . . . . 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 32317 . . . . . 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 1241 . . . . 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 214 . . . 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 215 . . 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 831 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   Latclat 15889   Atomscatm 32245   AtLatcal 32246   CvLatclc 32247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304
This theorem is referenced by:  cvlsupr3  32326  cvlsupr4  32327  cvlsupr5  32328  cvlsupr6  32329  4atexlemex2  33052  4atex  33057  4atex3  33062  cdleme02N  33204  cdleme0ex2N  33206  cdleme0moN  33207  cdleme0nex  33272
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