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Theorem cdleme3b 30711
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30718 and cdleme3 30719. (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme3b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  =/=  R
)

Proof of Theorem cdleme3b
StepHypRef Expression
1 simpll 731 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
2 simpr3l 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A
)
3 eqid 2404 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
4 cdleme1.a . . . . 5  |-  A  =  ( Atoms `  K )
53, 4atbase 29772 . . . 4  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
62, 5syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  (
Base `  K )
)
7 hllat 29846 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
87ad2antrr 707 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  Lat )
9 cdleme1.f . . . . 5  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
10 cdleme1.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
11 cdleme1.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
12 cdleme1.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
13 cdleme1.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
14 cdleme1.u . . . . . . . . . 10  |-  U  =  ( ( P  .\/  Q )  ./\  W )
1510, 11, 12, 4, 13, 14lhpat2 30527 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
16153adant3r3 1164 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  A
)
173, 4atbase 29772 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
1816, 17syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  (
Base `  K )
)
193, 11latjcl 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( R  .\/  U )  e.  ( Base `  K
) )
208, 6, 18, 19syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  U )  e.  ( Base `  K ) )
21 simpr2l 1016 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  A
)
223, 4atbase 29772 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2321, 22syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  (
Base `  K )
)
24 simpr1l 1014 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A
)
253, 4atbase 29772 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2624, 25syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  (
Base `  K )
)
273, 11latjcl 14434 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
288, 26, 6, 27syl3anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  R )  e.  ( Base `  K ) )
293, 13lhpbase 30480 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3029ad2antlr 708 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  W  e.  (
Base `  K )
)
313, 12latmcl 14435 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  e.  ( Base `  K ) )
328, 28, 30, 31syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P 
.\/  R )  ./\  W )  e.  ( Base `  K ) )
333, 11latjcl 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  R
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  (
Base `  K )
)
348, 23, 32, 33syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
)  e.  ( Base `  K ) )
353, 12latmcl 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
)  e.  ( Base `  K ) )  -> 
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  e.  (
Base `  K )
)
368, 20, 34, 35syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )  e.  ( Base `  K ) )
379, 36syl5eqel 2488 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  e.  (
Base `  K )
)
383, 11latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  F  e.  ( Base `  K
) )  ->  ( R  .\/  F )  e.  ( Base `  K
) )
398, 6, 37, 38syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  e.  ( Base `  K ) )
403, 11latjcl 14434 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
418, 26, 23, 40syl3anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K ) )
423, 10, 12latmle2 14461 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
438, 41, 30, 42syl3anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P 
.\/  Q )  ./\  W )  .<_  W )
4414, 43syl5eqbr 4205 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  .<_  W )
45 simpr3r 1019 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  -.  R  .<_  W )
46 nbrne2 4190 . . . . . . 7  |-  ( ( U  .<_  W  /\  -.  R  .<_  W )  ->  U  =/=  R
)
4744, 45, 46syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  =/=  R
)
4847necomd 2650 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  =/=  U
)
49 eqid 2404 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5011, 49, 4atcvr1 29899 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  =/=  U  <->  R (  <o  `  K )
( R  .\/  U
) ) )
511, 2, 16, 50syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  =/= 
U  <->  R (  <o  `  K
) ( R  .\/  U ) ) )
5248, 51mpbid 202 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R (  <o  `  K ) ( R 
.\/  U ) )
53 simpr3 965 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5424, 21, 533jca 1134 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )
5510, 11, 12, 4, 13, 14, 9cdleme1 30709 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R  .\/  U ) )
5654, 55syldan 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R 
.\/  U ) )
5752, 56breqtrrd 4198 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R (  <o  `  K ) ( R 
.\/  F ) )
583, 49cvrne 29764 . . 3  |-  ( ( ( K  e.  HL  /\  R  e.  ( Base `  K )  /\  ( R  .\/  F )  e.  ( Base `  K
) )  /\  R
(  <o  `  K )
( R  .\/  F
) )  ->  R  =/=  ( R  .\/  F
) )
591, 6, 39, 57, 58syl31anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  =/=  ( R  .\/  F ) )
60 oveq2 6048 . . . . . 6  |-  ( F  =  R  ->  ( R  .\/  F )  =  ( R  .\/  R
) )
6160adantl 453 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  /\  F  =  R )  ->  ( R  .\/  F )  =  ( R  .\/  R
) )
6211, 4hlatjidm 29851 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
631, 2, 62syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  R )  =  R )
6463adantr 452 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  /\  F  =  R )  ->  ( R  .\/  R )  =  R )
6561, 64eqtr2d 2437 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  /\  F  =  R )  ->  R  =  ( R  .\/  F ) )
6665ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( F  =  R  ->  R  =  ( R  .\/  F ) ) )
6766necon3d 2605 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  =/=  ( R  .\/  F
)  ->  F  =/=  R ) )
6859, 67mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  =/=  R
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429    <o ccvr 29745   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme36m  30943
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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