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Theorem cvlsupr2 28906
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 960 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  =/=  Q )
21necomd 2529 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  =/=  P )
3 simplr 731 . . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( P  .\/  R )  =  ( P  .\/  P
) )
5 oveq2 5866 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( Q  .\/  R )  =  ( Q  .\/  P
) )
64, 5eqeq12d 2297 . . . . . . . . . . 11  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  P )  =  ( Q  .\/  P ) ) )
7 eqcom 2285 . . . . . . . . . . 11  |-  ( ( P  .\/  P )  =  ( Q  .\/  P )  <->  ( Q  .\/  P )  =  ( P 
.\/  P ) )
86, 7syl6bb 252 . . . . . . . . . 10  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( Q  .\/  P )  =  ( P  .\/  P ) ) )
98adantl 452 . . . . . . . . 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 201 . . . . . . . 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 958 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  CvLat )
12 cvllat 28889 . . . . . . . . . . 11  |-  ( K  e.  CvLat  ->  K  e.  Lat )
1311, 12syl 15 . . . . . . . . . 10  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  Lat )
14 simpl21 1033 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  e.  A )
15 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
16 cvlsupr2.a . . . . . . . . . . . 12  |-  A  =  ( Atoms `  K )
1715, 16atbase 28852 . . . . . . . . . . 11  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1814, 17syl 15 . . . . . . . . . 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 14180 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P  .\/  P
)  =  P )
2113, 18, 20syl2anc 642 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .\/  P )  =  P )
2221adantr 451 . . . . . . . 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 2315 . . . . . . 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 423 . . . . . 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 1034 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  e.  A )
2615, 16atbase 28852 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2725, 26syl 15 . . . . . . . 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 14169 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( Q  .<_  P  <->  ( Q  .\/  P )  =  P ) )
3013, 27, 18, 29syl3anc 1182 . . . . . . 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 28888 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
3211, 31syl 15 . . . . . . . 8  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  AtLat )
3328, 16atcmp 28874 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Q  e.  A  /\  P  e.  A )  ->  ( Q  .<_  P  <->  Q  =  P ) )
3432, 25, 14, 33syl3anc 1182 . . . . . . 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 246 . . . . . 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 205 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  P  ->  Q  =  P ) )
3736necon3d 2484 . . . 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 14 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  =/=  P )
39 simplr 731 . . . . . . . . 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 5866 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( P  .\/  R )  =  ( P  .\/  Q
) )
41 oveq2 5866 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( Q  .\/  R )  =  ( Q  .\/  Q
) )
4240, 41eqeq12d 2297 . . . . . . . . . 10  |-  ( R  =  Q  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  Q )  =  ( Q  .\/  Q ) ) )
4342adantl 452 . . . . . . . . 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 201 . . . . . . . 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 14180 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  Q
)  =  Q )
4613, 27, 45syl2anc 642 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .\/  Q )  =  Q )
4746adantr 451 . . . . . . . 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 2315 . . . . . . 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 423 . . . . . 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 14169 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .<_  Q  <->  ( P  .\/  Q )  =  Q ) )
5113, 18, 27, 50syl3anc 1182 . . . . . . 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 28874 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )
5332, 14, 25, 52syl3anc 1182 . . . . . . 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 246 . . . . . 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 205 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( R  =  Q  ->  P  =  Q ) )
5655necon3d 2484 . . . 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 14 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  =/=  Q )
58 simpl23 1035 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  e.  A )
5915, 16atbase 28852 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
6058, 59syl 15 . . . . . 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 14166 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  R
) )
6213, 27, 60, 61syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  .<_  ( Q  .\/  R
) )
63 simpr 447 . . . . 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 4049 . . . 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 28899 . . . . 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 1195 . . . 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 14 . . 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 1132 . 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 963 . . 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 958 . . . . . . 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 15 . . . . . 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 1033 . . . . . . 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 15 . . . . . 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 1034 . . . . . . 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 15 . . . . . 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 14165 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
7771, 73, 75, 76syl3anc 1182 . . . . 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 4035 . . . 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 1035 . . . . . 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 962 . . . . . 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 28899 . . . . . 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 1195 . . . . 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 961 . . . . . . 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 2529 . . . . . 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 28898 . . . . . 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 1195 . . . . 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 205 . . . 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 206 . . 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 14 . 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 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 28826   AtLatcal 28827   CvLatclc 28828
This theorem is referenced by:  cvlsupr3  28907  cvlsupr4  28908  cvlsupr5  28909  cvlsupr6  28910  4atexlemex2  29633  4atex  29638  4atex3  29643  cdleme02N  29784  cdleme0ex2N  29786  cdleme0moN  29787  cdleme0nex  29852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885
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