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

Theorem diaocN 35052
Description: Value of partial isomorphism A at lattice orthocomplement (using a Sasaki projection to get orthocomplement relative to the fiducial co-atom  W). (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diaoc.j  |-  .\/  =  ( join `  K )
diaoc.m  |-  ./\  =  ( meet `  K )
diaoc.o  |-  ._|_  =  ( oc `  K )
diaoc.h  |-  H  =  ( LHyp `  K
)
diaoc.t  |-  T  =  ( ( LTrn `  K
) `  W )
diaoc.i  |-  I  =  ( ( DIsoA `  K
) `  W )
diaoc.n  |-  N  =  ( ( ocA `  K
) `  W )
Assertion
Ref Expression
diaocN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  ( (
(  ._|_  `  X )  .\/  (  ._|_  `  W
) )  ./\  W
) )  =  ( N `  ( I `
 X ) ) )

Proof of Theorem diaocN
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2450 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 diaoc.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 diaoc.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
52, 3, 4diadmclN 34964 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  e.  ( Base `  K
) )
6 eqid 2450 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
76, 3, 4diadmleN 34965 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X
( le `  K
) W )
8 diaoc.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
92, 6, 3, 8, 4diass 34969 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ( Base `  K
)  /\  X ( le `  K ) W ) )  ->  (
I `  X )  C_  T )
101, 5, 7, 9syl12anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  C_  T )
11 diaoc.j . . . 4  |-  .\/  =  ( join `  K )
12 diaoc.m . . . 4  |-  ./\  =  ( meet `  K )
13 diaoc.o . . . 4  |-  ._|_  =  ( oc `  K )
14 diaoc.n . . . 4  |-  N  =  ( ( ocA `  K
) `  W )
1511, 12, 13, 3, 8, 4, 14docavalN 35050 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( I `  X )  C_  T
)  ->  ( N `  ( I `  X
) )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
1610, 15syldan 470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( N `  ( I `  X ) )  =  ( I `  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
173, 4diaclN 34977 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  e.  ran  I )
18 intmin 4232 . . . . . . . . 9  |-  ( ( I `  X )  e.  ran  I  ->  |^| { z  e.  ran  I  |  ( I `  X )  C_  z }  =  ( I `  X ) )
1917, 18syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }  =  ( I `  X ) )
2019fveq2d 5779 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
)  =  ( `' I `  ( I `
 X ) ) )
213, 4diaf11N 34976 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
22 f1ocnvfv1 6068 . . . . . . . 8  |-  ( ( I : dom  I -1-1-onto-> ran  I  /\  X  e.  dom  I )  ->  ( `' I `  ( I `
 X ) )  =  X )
2321, 22sylan 471 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( `' I `  ( I `
 X ) )  =  X )
2420, 23eqtrd 2490 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
)  =  X )
2524fveq2d 5779 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (  ._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  ( I `  X ) 
C_  z } ) )  =  (  ._|_  `  X ) )
2625oveq1d 6191 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
) )  .\/  (  ._|_  `  W ) )  =  ( (  ._|_  `  X )  .\/  (  ._|_  `  W ) ) )
2726oveq1d 6191 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W )  =  ( ( (  ._|_  `  X )  .\/  (  ._|_  `  W ) ) 
./\  W ) )
2827fveq2d 5779 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  ( (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
) )  .\/  (  ._|_  `  W ) ) 
./\  W ) )  =  ( I `  ( ( (  ._|_  `  X )  .\/  (  ._|_  `  W ) ) 
./\  W ) ) )
2916, 28eqtr2d 2491 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  ( (
(  ._|_  `  X )  .\/  (  ._|_  `  W
) )  ./\  W
) )  =  ( N `  ( I `
 X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   {crab 2796    C_ wss 3412   |^|cint 4212   class class class wbr 4376   `'ccnv 4923   dom cdm 4924   ran crn 4925   -1-1-onto->wf1o 5501   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   occoc 14334   joincjn 15202   meetcmee 15203   HLchlt 33277   LHypclh 33910   LTrncltrn 34027   DIsoAcdia 34955   ocAcocaN 35046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-riotaBAD 32886
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-undef 6878  df-map 7302  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425  df-lvols 33426  df-lines 33427  df-psubsp 33429  df-pmap 33430  df-padd 33722  df-lhyp 33914  df-laut 33915  df-ldil 34030  df-ltrn 34031  df-trl 34085  df-disoa 34956  df-docaN 35047
This theorem is referenced by:  doca2N  35053  djajN  35064
  Copyright terms: Public domain W3C validator