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Theorem dochval 37514
Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
Hypotheses
Ref Expression
dochval.b 𝐵 = (Base‘𝐾)
dochval.g 𝐺 = (glb‘𝐾)
dochval.o = (oc‘𝐾)
dochval.h 𝐻 = (LHyp‘𝐾)
dochval.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dochval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochval.v 𝑉 = (Base‘𝑈)
dochval.n 𝑁 = ((ocH‘𝐾)‘𝑊)
Assertion
Ref Expression
dochval (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
Distinct variable groups:   𝑦,𝐵   𝑦,𝐾   𝑦,𝑊   𝑦,𝑋
Allowed substitution hints:   𝑈(𝑦)   𝐺(𝑦)   𝐻(𝑦)   𝐼(𝑦)   𝑁(𝑦)   (𝑦)   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem dochval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dochval.b . . . . 5 𝐵 = (Base‘𝐾)
2 dochval.g . . . . 5 𝐺 = (glb‘𝐾)
3 dochval.o . . . . 5 = (oc‘𝐾)
4 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
5 dochval.i . . . . 5 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6 dochval.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
7 dochval.v . . . . 5 𝑉 = (Base‘𝑈)
8 dochval.n . . . . 5 𝑁 = ((ocH‘𝐾)‘𝑊)
91, 2, 3, 4, 5, 6, 7, 8dochfval 37513 . . . 4 ((𝐾𝑌𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
109adantr 474 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
1110fveq1d 6450 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋))
127fvexi 6462 . . . . . 6 𝑉 ∈ V
1312elpw2 5064 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1413biimpri 220 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1514adantl 475 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑋 ∈ 𝒫 𝑉)
16 fvex 6461 . . 3 (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V
17 sseq1 3845 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼𝑦) ↔ 𝑋 ⊆ (𝐼𝑦)))
1817rabbidv 3386 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝐵𝑥 ⊆ (𝐼𝑦)} = {𝑦𝐵𝑋 ⊆ (𝐼𝑦)})
1918fveq2d 6452 . . . . . 6 (𝑥 = 𝑋 → (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}) = (𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))
2019fveq2d 6452 . . . . 5 (𝑥 = 𝑋 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)})))
2120fveq2d 6452 . . . 4 (𝑥 = 𝑋 → (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
22 eqid 2778 . . . 4 (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))
2321, 22fvmptg 6542 . . 3 ((𝑋 ∈ 𝒫 𝑉 ∧ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
2415, 16, 23sylancl 580 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
2511, 24eqtrd 2814 1 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  wss 3792  𝒫 cpw 4379  cmpt 4967  cfv 6137  Basecbs 16266  occoc 16357  glbcglb 17340  LHypclh 36147  DVecHcdvh 37241  DIsoHcdih 37391  ocHcoch 37510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-doch 37511
This theorem is referenced by:  dochval2  37515  dochcl  37516  dochvalr  37520  dochss  37528
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