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Theorem cvlsupr2 30068
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 2681 . . . 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 6081 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( P  .\/  R )  =  ( P  .\/  P
) )
5 oveq2 6081 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( Q  .\/  R )  =  ( Q  .\/  P
) )
64, 5eqeq12d 2449 . . . . . . . . . . 11  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  P )  =  ( Q  .\/  P ) ) )
7 eqcom 2437 . . . . . . . . . . 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 30051 . . . . . . . . . . 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 2435 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
16 cvlsupr2.a . . . . . . . . . . . 12  |-  A  =  ( Atoms `  K )
1715, 16atbase 30014 . . . . . . . . . . 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 14495 . . . . . . . . . 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 2467 . . . . . . 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 30014 . . . . . . . . 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 14484 . . . . . . . 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 30050 . . . . . . . . 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 30036 . . . . . . . 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 2636 . . . 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 6081 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( P  .\/  R )  =  ( P  .\/  Q
) )
41 oveq2 6081 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( Q  .\/  R )  =  ( Q  .\/  Q
) )
4240, 41eqeq12d 2449 . . . . . . . . . 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 14495 . . . . . . . . . 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 2467 . . . . . . 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 14484 . . . . . . . 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 30036 . . . . . . . 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 2636 . . . 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 30014 . . . . . . 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 14481 . . . . . 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 4230 . . . 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 30061 . . . . 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 14480 . . . . . 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 4216 . . . 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 30061 . . . . . 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 2681 . . . . . 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 30060 . . . . . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29988   AtLatcal 29989   CvLatclc 29990
This theorem is referenced by:  cvlsupr3  30069  cvlsupr4  30070  cvlsupr5  30071  cvlsupr6  30072  4atexlemex2  30795  4atex  30800  4atex3  30805  cdleme02N  30946  cdleme0ex2N  30948  cdleme0moN  30949  cdleme0nex  31014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-join 14425  df-lat 14467  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047
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