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Theorem doch2val2 30927
Description: Double orthocomplement for  DVecH vector space. (Contributed by NM, 26-Jul-2014.)
Hypotheses
Ref Expression
doch2val2.h  |-  H  =  ( LHyp `  K
)
doch2val2.i  |-  I  =  ( ( DIsoH `  K
) `  W )
doch2val2.u  |-  U  =  ( ( DVecH `  K
) `  W )
doch2val2.v  |-  V  =  ( Base `  U
)
doch2val2.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
doch2val2.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
doch2val2.x  |-  ( ph  ->  X  C_  V )
Assertion
Ref Expression
doch2val2  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  X ) )  = 
|^| { z  e.  ran  I  |  X  C_  z } )
Distinct variable groups:    z, H    z, I    z, K    z, V    z, W    z, X
Allowed substitution hints:    ph( z)    U( z)   
._|_ ( z)

Proof of Theorem doch2val2
StepHypRef Expression
1 doch2val2.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 doch2val2.x . . . 4  |-  ( ph  ->  X  C_  V )
3 eqid 2283 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
4 doch2val2.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 doch2val2.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
6 doch2val2.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
7 doch2val2.v . . . . 5  |-  V  =  ( Base `  U
)
8 doch2val2.o . . . . 5  |-  ._|_  =  ( ( ocH `  K
) `  W )
93, 4, 5, 6, 7, 8dochval2 30915 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  =  ( I `  ( ( oc `  K ) `
 ( `' I `  |^| { z  e. 
ran  I  |  X  C_  z } ) ) ) )
101, 2, 9syl2anc 642 . . 3  |-  ( ph  ->  (  ._|_  `  X )  =  ( I `  ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) )
1110fveq2d 5529 . 2  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  X ) )  =  (  ._|_  `  ( I `
 ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) ) )
121simpld 445 . . . . 5  |-  ( ph  ->  K  e.  HL )
13 hlop 28925 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
1412, 13syl 15 . . . 4  |-  ( ph  ->  K  e.  OP )
15 ssrab2 3258 . . . . . . 7  |-  { z  e.  ran  I  |  X  C_  z }  C_ 
ran  I
1615a1i 10 . . . . . 6  |-  ( ph  ->  { z  e.  ran  I  |  X  C_  z }  C_  ran  I )
174, 5, 6, 7dih1rn 30850 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  V  e.  ran  I
)
181, 17syl 15 . . . . . . . 8  |-  ( ph  ->  V  e.  ran  I
)
19 sseq2 3200 . . . . . . . . 9  |-  ( z  =  V  ->  ( X  C_  z  <->  X  C_  V
) )
2019elrab 2923 . . . . . . . 8  |-  ( V  e.  { z  e. 
ran  I  |  X  C_  z }  <->  ( V  e.  ran  I  /\  X  C_  V ) )
2118, 2, 20sylanbrc 645 . . . . . . 7  |-  ( ph  ->  V  e.  { z  e.  ran  I  |  X  C_  z }
)
22 ne0i 3461 . . . . . . 7  |-  ( V  e.  { z  e. 
ran  I  |  X  C_  z }  ->  { z  e.  ran  I  |  X  C_  z }  =/=  (/) )
2321, 22syl 15 . . . . . 6  |-  ( ph  ->  { z  e.  ran  I  |  X  C_  z }  =/=  (/) )
244, 5dihintcl 30907 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( { z  e.  ran  I  |  X  C_  z }  C_ 
ran  I  /\  {
z  e.  ran  I  |  X  C_  z }  =/=  (/) ) )  ->  |^| { z  e.  ran  I  |  X  C_  z }  e.  ran  I )
251, 16, 23, 24syl12anc 1180 . . . . 5  |-  ( ph  ->  |^| { z  e. 
ran  I  |  X  C_  z }  e.  ran  I )
26 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2726, 4, 5dihcnvcl 30834 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  |^| { z  e. 
ran  I  |  X  C_  z }  e.  ran  I )  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  ( Base `  K ) )
281, 25, 27syl2anc 642 . . . 4  |-  ( ph  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } )  e.  (
Base `  K )
)
2926, 3opoccl 28757 . . . 4  |-  ( ( K  e.  OP  /\  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  e.  (
Base `  K )
)
3014, 28, 29syl2anc 642 . . 3  |-  ( ph  ->  ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  e.  (
Base `  K )
)
3126, 3, 4, 5, 8dochvalr2 30925 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  e.  (
Base `  K )
)  ->  (  ._|_  `  ( I `  (
( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) )  =  ( I `  ( ( oc `  K ) `  (
( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) ) )
321, 30, 31syl2anc 642 . 2  |-  ( ph  ->  (  ._|_  `  ( I `
 ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) )  =  ( I `  ( ( oc `  K ) `  (
( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) ) )
3326, 3opococ 28758 . . . . 5  |-  ( ( K  e.  OP  /\  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) )  =  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )
3414, 28, 33syl2anc 642 . . . 4  |-  ( ph  ->  ( ( oc `  K ) `  (
( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) )  =  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) )
3534fveq2d 5529 . . 3  |-  ( ph  ->  ( I `  (
( oc `  K
) `  ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) )  =  ( I `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) )
364, 5dihcnvid2 30836 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  |^| { z  e. 
ran  I  |  X  C_  z }  e.  ran  I )  ->  (
I `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  =  |^| { z  e.  ran  I  |  X  C_  z } )
371, 25, 36syl2anc 642 . . 3  |-  ( ph  ->  ( I `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  =  |^| { z  e.  ran  I  |  X  C_  z } )
3835, 37eqtrd 2315 . 2  |-  ( ph  ->  ( I `  (
( oc `  K
) `  ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ) )  =  |^| { z  e.  ran  I  |  X  C_  z }
)
3911, 32, 383eqtrd 2319 1  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  X ) )  = 
|^| { z  e.  ran  I  |  X  C_  z } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   |^|cint 3862   `'ccnv 4688   ran crn 4690   ` cfv 5255   Basecbs 13148   occoc 13216   OPcops 28735   HLchlt 28913   LHypclh 29546   DVecHcdvh 30641   DIsoHcdih 30791   ocHcoch 30910
This theorem is referenced by:  dochspss  30941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 28539  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061  df-lvols 29062  df-lines 29063  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667  df-trl 29721  df-tendo 30317  df-edring 30319  df-disoa 30592  df-dvech 30642  df-dib 30702  df-dic 30736  df-dih 30792  df-doch 30911
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