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Theorem docavalN 36831
Description: Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
docaval.j = (join‘𝐾)
docaval.m = (meet‘𝐾)
docaval.o = (oc‘𝐾)
docaval.h 𝐻 = (LHyp‘𝐾)
docaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
docaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
docaval.n 𝑁 = ((ocA‘𝐾)‘𝑊)
Assertion
Ref Expression
docavalN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
Distinct variable groups:   𝑧,𝐾   𝑧,𝐼   𝑧,𝑊   𝑧,𝑇   𝑧,𝑋
Allowed substitution hints:   𝐻(𝑧)   (𝑧)   (𝑧)   𝑁(𝑧)   (𝑧)

Proof of Theorem docavalN
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 docaval.j . . . . 5 = (join‘𝐾)
2 docaval.m . . . . 5 = (meet‘𝐾)
3 docaval.o . . . . 5 = (oc‘𝐾)
4 docaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
5 docaval.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 docaval.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7 docaval.n . . . . 5 𝑁 = ((ocA‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7docafvalN 36830 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
98adantr 472 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
109fveq1d 6306 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))‘𝑋))
11 fvex 6314 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) ∈ V
125, 11eqeltri 2799 . . . . . 6 𝑇 ∈ V
1312elpw2 4933 . . . . 5 (𝑋 ∈ 𝒫 𝑇𝑋𝑇)
1413biimpri 218 . . . 4 (𝑋𝑇𝑋 ∈ 𝒫 𝑇)
1514adantl 473 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑋 ∈ 𝒫 𝑇)
16 fvex 6314 . . 3 (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)) ∈ V
17 sseq1 3732 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥𝑧𝑋𝑧))
1817rabbidv 3293 . . . . . . . . . 10 (𝑥 = 𝑋 → {𝑧 ∈ ran 𝐼𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑋𝑧})
1918inteqd 4588 . . . . . . . . 9 (𝑥 = 𝑋 {𝑧 ∈ ran 𝐼𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑋𝑧})
2019fveq2d 6308 . . . . . . . 8 (𝑥 = 𝑋 → (𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧}) = (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))
2120fveq2d 6308 . . . . . . 7 (𝑥 = 𝑋 → ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) = ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})))
2221oveq1d 6780 . . . . . 6 (𝑥 = 𝑋 → (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) = (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)))
2322oveq1d 6780 . . . . 5 (𝑥 = 𝑋 → ((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊) = ((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊))
2423fveq2d 6308 . . . 4 (𝑥 = 𝑋 → (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
25 eqid 2724 . . . 4 (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))
2624, 25fvmptg 6394 . . 3 ((𝑋 ∈ 𝒫 𝑇 ∧ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)) ∈ V) → ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))‘𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
2715, 16, 26sylancl 697 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))‘𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
2810, 27eqtrd 2758 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  {crab 3018  Vcvv 3304  wss 3680  𝒫 cpw 4266   cint 4583  cmpt 4837  ccnv 5217  ran crn 5219  cfv 6001  (class class class)co 6765  occoc 16072  joincjn 17066  meetcmee 17067  HLchlt 35057  LHypclh 35690  LTrncltrn 35807  DIsoAcdia 36736  ocAcocaN 36827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-docaN 36828
This theorem is referenced by:  docaclN  36832  diaocN  36833
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