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Theorem cdlemc3 29650
Description: Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l  |-  .<_  =  ( le `  K )
cdlemc3.j  |-  .\/  =  ( join `  K )
cdlemc3.m  |-  ./\  =  ( meet `  K )
cdlemc3.a  |-  A  =  ( Atoms `  K )
cdlemc3.h  |-  H  =  ( LHyp `  K
)
cdlemc3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemc3.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemc3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .<_  ( Q  .\/  ( R `  F ) )  ->  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )

Proof of Theorem cdlemc3
StepHypRef Expression
1 simpll 732 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  HL )
2 simpl 445 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpr1 963 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  F  e.  T )
4 simpr2l 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  P  e.  A )
5 cdlemc3.l . . . . 5  |-  .<_  =  ( le `  K )
6 cdlemc3.a . . . . 5  |-  A  =  ( Atoms `  K )
7 cdlemc3.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 cdlemc3.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnat 29597 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
102, 3, 4, 9syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  P )  e.  A )
11 simpr3l 1018 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  e.  A )
12 eqid 2285 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
13 cdlemc3.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
1412, 7, 8, 13trlcl 29621 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
153, 14syldan 458 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( R `  F )  e.  ( Base `  K
) )
165, 6, 7, 8ltrnel 29596 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
17163adant3r3 1164 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  e.  A  /\  -.  ( F `  P
)  .<_  W ) )
185, 6, 7, 8, 13trlnle 29643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  ->  -.  ( F `  P )  .<_  ( R `  F
) )
192, 3, 17, 18syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  -.  ( F `  P ) 
.<_  ( R `  F
) )
20 cdlemc3.j . . . 4  |-  .\/  =  ( join `  K )
2112, 5, 20, 6hlexch2 28840 . . 3  |-  ( ( K  e.  HL  /\  ( ( F `  P )  e.  A  /\  Q  e.  A  /\  ( R `  F
)  e.  ( Base `  K ) )  /\  -.  ( F `  P
)  .<_  ( R `  F ) )  -> 
( ( F `  P )  .<_  ( Q 
.\/  ( R `  F ) )  ->  Q  .<_  ( ( F `
 P )  .\/  ( R `  F ) ) ) )
221, 10, 11, 15, 19, 21syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .<_  ( Q  .\/  ( R `  F ) )  ->  Q  .<_  ( ( F `  P
)  .\/  ( R `  F ) ) ) )
235, 20, 6, 7, 8, 13trljat2 29624 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
24233adant3r3 1164 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .\/  ( R `  F ) )  =  ( P  .\/  ( F `  P )
) )
2524breq2d 4037 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .<_  ( ( F `
 P )  .\/  ( R `  F ) )  <->  Q  .<_  ( P 
.\/  ( F `  P ) ) ) )
2622, 25sylibd 207 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .<_  ( Q  .\/  ( R `  F ) )  ->  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Atomscatm 28721   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   trLctrl 29615
This theorem is referenced by:  cdlemc4  29651
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616
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