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Theorem docavalN 35430
 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 35429 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
98adantr 480 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
109fveq1d 6105 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))‘𝑋))
11 fvex 6113 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) ∈ V
125, 11eqeltri 2684 . . . . . 6 𝑇 ∈ V
1312elpw2 4755 . . . . 5 (𝑋 ∈ 𝒫 𝑇𝑋𝑇)
1413biimpri 217 . . . 4 (𝑋𝑇𝑋 ∈ 𝒫 𝑇)
1514adantl 481 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑋 ∈ 𝒫 𝑇)
16 fvex 6113 . . 3 (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)) ∈ V
17 sseq1 3589 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥𝑧𝑋𝑧))
1817rabbidv 3164 . . . . . . . . . 10 (𝑥 = 𝑋 → {𝑧 ∈ ran 𝐼𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑋𝑧})
1918inteqd 4415 . . . . . . . . 9 (𝑥 = 𝑋 {𝑧 ∈ ran 𝐼𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑋𝑧})
2019fveq2d 6107 . . . . . . . 8 (𝑥 = 𝑋 → (𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧}) = (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))
2120fveq2d 6107 . . . . . . 7 (𝑥 = 𝑋 → ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) = ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})))
2221oveq1d 6564 . . . . . 6 (𝑥 = 𝑋 → (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) = (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)))
2322oveq1d 6564 . . . . 5 (𝑥 = 𝑋 → ((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊) = ((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊))
2423fveq2d 6107 . . . 4 (𝑥 = 𝑋 → (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))
25 eqid 2610 . . . 4 (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))
2624, 25fvmptg 6189 . . 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 693 . 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 2644 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 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∩ cint 4410   ↦ cmpt 4643  ◡ccnv 5037  ran crn 5039  ‘cfv 5804  (class class class)co 6549  occoc 15776  joincjn 16767  meetcmee 16768  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  DIsoAcdia 35335  ocAcocaN 35426 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-docaN 35427 This theorem is referenced by:  docaclN  35431  diaocN  35432
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