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Theorem cvlsupr2 34140
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 1001 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  =/=  Q )
21necomd 2738 . . . 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 6290 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( P  .\/  R )  =  ( P  .\/  P
) )
5 oveq2 6290 . . . . . . . . . . . 12  |-  ( R  =  P  ->  ( Q  .\/  R )  =  ( Q  .\/  P
) )
64, 5eqeq12d 2489 . . . . . . . . . . 11  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  P )  =  ( Q  .\/  P ) ) )
7 eqcom 2476 . . . . . . . . . . 11  |-  ( ( P  .\/  P )  =  ( Q  .\/  P )  <->  ( Q  .\/  P )  =  ( P 
.\/  P ) )
86, 7syl6bb 261 . . . . . . . . . 10  |-  ( R  =  P  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( Q  .\/  P )  =  ( P  .\/  P ) ) )
98adantl 466 . . . . . . . . 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 999 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  K  e.  CvLat )
12 cvllat 34123 . . . . . . . . . . 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 1074 . . . . . . . . . . 11  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  P  e.  A )
15 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
16 cvlsupr2.a . . . . . . . . . . . 12  |-  A  =  ( Atoms `  K )
1715, 16atbase 34086 . . . . . . . . . . 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 15557 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P  .\/  P
)  =  P )
2113, 18, 20syl2anc 661 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( P  .\/  P )  =  P )
2221adantr 465 . . . . . . . 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 2508 . . . . . . 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 434 . . . . . 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 1075 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  Q  e.  A )
2615, 16atbase 34086 . . . . . . . . 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 15546 . . . . . . . 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 34122 . . . . . . . . 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 34108 . . . . . . . 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 2691 . . . 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 6290 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( P  .\/  R )  =  ( P  .\/  Q
) )
41 oveq2 6290 . . . . . . . . . . 11  |-  ( R  =  Q  ->  ( Q  .\/  R )  =  ( Q  .\/  Q
) )
4240, 41eqeq12d 2489 . . . . . . . . . 10  |-  ( R  =  Q  ->  (
( P  .\/  R
)  =  ( Q 
.\/  R )  <->  ( P  .\/  Q )  =  ( Q  .\/  Q ) ) )
4342adantl 466 . . . . . . . . 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 15557 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  Q
)  =  Q )
4613, 27, 45syl2anc 661 . . . . . . . . 9  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  ( Q  .\/  Q )  =  Q )
4746adantr 465 . . . . . . . 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 2508 . . . . . . 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 434 . . . . . 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 15546 . . . . . . . 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 34108 . . . . . . . 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 2691 . . . 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 1076 . . . . . . 7  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  Q )  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) )  ->  R  e.  A )
5915, 16atbase 34086 . . . . . . 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 15543 . . . . . 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 461 . . . . 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 4473 . . . 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 34133 . . . . 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 1176 . 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 1004 . . 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 999 . . . . . . 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 1074 . . . . . . 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 1075 . . . . . . 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 15542 . . . . . 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 4459 . . . 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 1076 . . . . . 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 1003 . . . . . 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 34133 . . . . . 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 1002 . . . . . . 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 2738 . . . . . 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 34132 . . . . . 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 830 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   Latclat 15528   Atomscatm 34060   AtLatcal 34061   CvLatclc 34062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119
This theorem is referenced by:  cvlsupr3  34141  cvlsupr4  34142  cvlsupr5  34143  cvlsupr6  34144  4atexlemex2  34867  4atex  34872  4atex3  34877  cdleme02N  35018  cdleme0ex2N  35020  cdleme0moN  35021  cdleme0nex  35086
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