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Theorem diaocN 37268
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 455 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 diaoc.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 diaoc.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
52, 3, 4diadmclN 37180 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  e.  ( Base `  K
) )
6 eqid 2454 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
76, 3, 4diadmleN 37181 . . . 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 37185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ( Base `  K
)  /\  X ( le `  K ) W ) )  ->  (
I `  X )  C_  T )
101, 5, 7, 9syl12anc 1224 . . 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 37266 . . 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 468 . 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 37193 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  e.  ran  I )
18 intmin 4291 . . . . . . . . 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 5852 . . . . . . 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 37192 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
22 f1ocnvfv1 6157 . . . . . . . 8  |-  ( ( I : dom  I -1-1-onto-> ran  I  /\  X  e.  dom  I )  ->  ( `' I `  ( I `
 X ) )  =  X )
2321, 22sylan 469 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( `' I `  ( I `
 X ) )  =  X )
2420, 23eqtrd 2495 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
)  =  X )
2524fveq2d 5852 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (  ._|_  `  ( `' I `  |^| { z  e. 
ran  I  |  ( I `  X ) 
C_  z } ) )  =  (  ._|_  `  X ) )
2625oveq1d 6285 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
(  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  ( I `  X
)  C_  z }
) )  .\/  (  ._|_  `  W ) )  =  ( (  ._|_  `  X )  .\/  (  ._|_  `  W ) ) )
2726oveq1d 6285 . . 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 5852 . 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 2496 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 367    = wceq 1398    e. wcel 1823   {crab 2808    C_ wss 3461   |^|cint 4271   class class class wbr 4439   `'ccnv 4987   dom cdm 4988   ran crn 4989   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   occoc 14795   joincjn 15775   meetcmee 15776   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   DIsoAcdia 37171   ocAcocaN 37262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35100
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300  df-disoa 37172  df-docaN 37263
This theorem is referenced by:  doca2N  37269  djajN  37280
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