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Theorem docavalN 38874
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 38873 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
98adantr 484 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
109fveq1d 6719 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))‘𝑋))
115fvexi 6731 . . . . . 6 𝑇 ∈ V
1211elpw2 5238 . . . . 5 (𝑋 ∈ 𝒫 𝑇𝑋𝑇)
1312biimpri 231 . . . 4 (𝑋𝑇𝑋 ∈ 𝒫 𝑇)
1413adantl 485 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑋 ∈ 𝒫 𝑇)
15 fvex 6730 . . 3 (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)) ∈ V
16 sseq1 3926 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥𝑧𝑋𝑧))
1716rabbidv 3390 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑧 ∈ ran 𝐼𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑋𝑧})
1817inteqd 4864 . . . . . . . 8 (𝑥 = 𝑋 {𝑧 ∈ ran 𝐼𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑋𝑧})
1918fveq2d 6721 . . . . . . 7 (𝑥 = 𝑋 → (𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧}) = (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))
2019fveq2d 6721 . . . . . 6 (𝑥 = 𝑋 → ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) = ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})))
2120oveq1d 7228 . . . . 5 (𝑥 = 𝑋 → (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) = (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)))
2221fvoveq1d 7235 . . . 4 (𝑥 = 𝑋 → (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
23 eqid 2737 . . . 4 (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))
2422, 23fvmptg 6816 . . 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)))
2514, 15, 24sylancl 589 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))‘𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
2610, 25eqtrd 2777 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  {crab 3065  Vcvv 3408  wss 3866  𝒫 cpw 4513   cint 4859  cmpt 5135  ccnv 5550  ran crn 5552  cfv 6380  (class class class)co 7213  occoc 16810  joincjn 17818  meetcmee 17819  HLchlt 37101  LHypclh 37735  LTrncltrn 37852  DIsoAcdia 38779  ocAcocaN 38870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-docaN 38871
This theorem is referenced by:  docaclN  38875  diaocN  38876
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