Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trljat2 Unicode version

Theorem trljat2 29623
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. (Contributed by NM, 25-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
trljat2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )

Proof of Theorem trljat2
StepHypRef Expression
1 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 trljat.l . . . . . 6  |-  .<_  =  ( le `  K )
3 trljat.a . . . . . 6  |-  A  =  ( Atoms `  K )
4 trljat.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 trljat.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
62, 3, 4, 5ltrnat 29596 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
763adant3r 1181 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  A
)
8 hllat 28820 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
91, 8syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
10 simp3l 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
11 eqid 2284 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3atbase 28746 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1310, 12syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
14 simp1 957 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp2 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
1611, 4, 5ltrncl 29581 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
1714, 15, 13, 16syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  (
Base `  K )
)
18 trljat.j . . . . . 6  |-  .\/  =  ( join `  K )
1911, 18latjcl 14150 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
209, 13, 17, 19syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
)
21 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
2211, 4lhpbase 29454 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2321, 22syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
2411, 2, 18latlej2 14161 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( F `  P )  .<_  ( P  .\/  ( F `  P )
) )
259, 13, 17, 24syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  .<_  ( P 
.\/  ( F `  P ) ) )
26 eqid 2284 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
2711, 2, 18, 26, 3atmod2i1 29317 . . . 4  |-  ( ( K  e.  HL  /\  ( ( F `  P )  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  ( F `  P )  .<_  ( P  .\/  ( F `  P )
) )  ->  (
( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  ( F `  P ) )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  ( F `  P ) ) ) )
281, 7, 20, 23, 25, 27syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( P  .\/  ( F `  P )
) ( meet `  K
) W )  .\/  ( F `  P ) )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  ( F `  P ) ) ) )
292, 3, 4, 5ltrnel 29595 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
30 eqid 2284 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
312, 18, 30, 3, 4lhpjat1 29476 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F `
 P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  -> 
( W  .\/  ( F `  P )
)  =  ( 1.
`  K ) )
321, 21, 29, 31syl21anc 1183 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  ( F `  P
) )  =  ( 1. `  K ) )
3332oveq2d 5835 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( W  .\/  ( F `  P )
) )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) ) )
34 hlol 28818 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
351, 34syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OL )
3611, 26, 30olm11 28684 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
) )  ->  (
( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
3735, 20, 36syl2anc 644 . . 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 ) ) )
3828, 33, 373eqtrrd 2321 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  ( F `  P ) ) )
39 trljat.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
402, 18, 26, 3, 4, 5, 39trlval2 29619 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) W ) )
4140oveq1d 5834 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( R `  F )  .\/  ( F `  P
) )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  ( F `  P ) ) )
4211, 4, 5, 39trlcl 29620 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
4314, 15, 42syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  e.  (
Base `  K )
)
4411, 18latjcom 14159 . . 3  |-  ( ( K  e.  Lat  /\  ( R `  F )  e.  ( Base `  K
)  /\  ( F `  P )  e.  (
Base `  K )
)  ->  ( ( R `  F )  .\/  ( F `  P
) )  =  ( ( F `  P
)  .\/  ( R `  F ) ) )
459, 43, 17, 44syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( R `  F )  .\/  ( F `  P
) )  =  ( ( F `  P
)  .\/  ( R `  F ) ) )
4638, 41, 453eqtr2rd 2323 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  .\/  ( R `  F
) )  =  ( 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 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   1.cp1 14138   Latclat 14145   OLcol 28631   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  trljat3  29624  cdlemc3  29649
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 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
  Copyright terms: Public domain W3C validator