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Theorem cdlemd3 30682
Description: Part of proof of Lemma D in [Crawley] p. 113. The  R  =/=  P requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd3.l  |-  .<_  =  ( le `  K )
cdlemd3.j  |-  .\/  =  ( join `  K )
cdlemd3.a  |-  A  =  ( Atoms `  K )
cdlemd3.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemd3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  S
) )

Proof of Theorem cdlemd3
StepHypRef Expression
1 simp33 995 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
2 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
3 simp31 993 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  A )
4 simp32 994 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  A )
5 simp21l 1074 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
6 simp233 1103 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  =/=  P )
7 cdlemd3.l . . . . 5  |-  .<_  =  ( le `  K )
8 cdlemd3.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdlemd3.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatexch1 29877 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  P  e.  A
)  /\  R  =/=  P )  ->  ( R  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  R ) ) )
112, 3, 4, 5, 6, 10syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  R ) ) )
12 simp22l 1076 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
137, 8, 9hlatlej1 29857 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
142, 5, 12, 13syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  ( P  .\/  Q ) )
15 simp232 1102 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q ) )
16 hllat 29846 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
172, 16syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
18 eqid 2404 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1918, 9atbase 29772 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
205, 19syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  ( Base `  K )
)
2118, 9atbase 29772 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
223, 21syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  ( Base `  K )
)
2318, 9atbase 29772 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2412, 23syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  ( Base `  K )
)
2518, 8latjcl 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2617, 20, 24, 25syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
2718, 7, 8latjle12 14446 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) )  <-> 
( P  .\/  R
)  .<_  ( P  .\/  Q ) ) )
2817, 20, 22, 26, 27syl13anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  <->  ( P  .\/  R )  .<_  ( P  .\/  Q ) ) )
2914, 15, 28mpbi2and 888 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  R )  .<_  ( P 
.\/  Q ) )
3018, 9atbase 29772 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
314, 30syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  ( Base `  K )
)
3218, 8latjcl 14434 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
3317, 20, 22, 32syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  R )  e.  (
Base `  K )
)
3418, 7lattr 14440 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( S  .<_  ( P 
.\/  R )  /\  ( P  .\/  R ) 
.<_  ( P  .\/  Q
) )  ->  S  .<_  ( P  .\/  Q
) ) )
3517, 31, 33, 26, 34syl13anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( S  .<_  ( P  .\/  R )  /\  ( P 
.\/  R )  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( P  .\/  Q ) ) )
3629, 35mpan2d 656 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( S  .<_  ( P  .\/  R
)  ->  S  .<_  ( P  .\/  Q ) ) )
3711, 36syld 42 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  Q ) ) )
381, 37mtod 170 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  S
) )
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   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdlemd4  30683
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-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-join 14388  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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