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Theorem trljat1 30648
Description: The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. Todo: shorten with atmod3i1 30346? (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
trljat.l  |-  .<_  =  ( le `  K )
trljat.j  |-  .\/  =  ( join `  K )
trljat.a  |-  A  =  ( Atoms `  K )
trljat.h  |-  H  =  ( LHyp `  K
)
trljat.t  |-  T  =  ( ( LTrn `  K
) `  W )
trljat.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trljat1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )

Proof of Theorem trljat1
StepHypRef Expression
1 trljat.l . . . 4  |-  .<_  =  ( le `  K )
2 trljat.j . . . 4  |-  .\/  =  ( join `  K )
3 eqid 2404 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
4 trljat.a . . . 4  |-  A  =  ( Atoms `  K )
5 trljat.h . . . 4  |-  H  =  ( LHyp `  K
)
6 trljat.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
7 trljat.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
81, 2, 3, 4, 5, 6, 7trlval2 30645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) W ) )
98oveq1d 6055 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( R `  F )  .\/  P )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P ) )
10 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
11 hllat 29846 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
13 simp3l 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
14 eqid 2404 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
1514, 4atbase 29772 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1613, 15syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
1714, 5, 6, 7trlcl 30646 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
18173adant3 977 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  e.  (
Base `  K )
)
1914, 2latjcom 14443 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( R `  F )  e.  ( Base `  K
) )  ->  ( P  .\/  ( R `  F ) )  =  ( ( R `  F )  .\/  P
) )
2012, 16, 18, 19syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( ( R `  F
)  .\/  P )
)
2114, 5, 6ltrncl 30607 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2216, 21syld3an3 1229 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2314, 2latjcl 14434 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
2412, 16, 22, 23syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
)
25 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
2614, 5lhpbase 30480 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2725, 26syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
2814, 1, 2latlej1 14444 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  ( F `  P )
) )
2912, 16, 22, 28syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  .<_  ( P  .\/  ( F `
 P ) ) )
3014, 1, 2, 3, 4atmod2i1 30343 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  ( F `  P )
) )  ->  (
( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  P ) ) )
3110, 13, 24, 27, 29, 30syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( P  .\/  ( F `  P )
) ( meet `  K
) W )  .\/  P )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  P ) ) )
32 eqid 2404 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
331, 2, 32, 4, 5lhpjat1 30502 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
34333adant2 976 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  ( 1. `  K ) )
3534oveq2d 6056 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( W  .\/  P
) )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) ) )
36 hlol 29844 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3710, 36syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OL )
3814, 3, 32olm11 29710 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
) )  ->  (
( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
3937, 24, 38syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( 1. `  K
) )  =  ( P  .\/  ( F `
 P ) ) )
4031, 35, 393eqtrrd 2441 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P ) )
419, 20, 403eqtr4d 2446 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   1.cp1 14422   Latclat 14429   OLcol 29657   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  trljat3  30650  trlval4  30670  trlval5  30671  cdlemc5  30677  cdlemk1  31313  cdlemk8  31320  cdlemki  31323  cdlemksv2  31329  cdlemk7  31330  cdlemk12  31332  cdlemk15  31337  cdlemk7u  31352  cdlemk12u  31354  cdlemk21N  31355  cdlemk20  31356  cdlemk22  31375  cdlemm10N  31601
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-map 6979  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  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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