Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dochval Structured version   Visualization version   GIF version

Theorem dochval 38605
 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 38604 . . . 4 ((𝐾𝑌𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
109adantr 484 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
1110fveq1d 6654 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋))
127fvexi 6666 . . . . . 6 𝑉 ∈ V
1312elpw2 5224 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1413biimpri 231 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1514adantl 485 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑋 ∈ 𝒫 𝑉)
16 fvex 6665 . . 3 (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V
17 sseq1 3967 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼𝑦) ↔ 𝑋 ⊆ (𝐼𝑦)))
1817rabbidv 3455 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝐵𝑥 ⊆ (𝐼𝑦)} = {𝑦𝐵𝑋 ⊆ (𝐼𝑦)})
1918fveq2d 6656 . . . . . 6 (𝑥 = 𝑋 → (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}) = (𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))
2019fveq2d 6656 . . . . 5 (𝑥 = 𝑋 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)})))
2120fveq2d 6656 . . . 4 (𝑥 = 𝑋 → (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
22 eqid 2822 . . . 4 (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))
2321, 22fvmptg 6748 . . 3 ((𝑋 ∈ 𝒫 𝑉 ∧ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
2415, 16, 23sylancl 589 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
2511, 24eqtrd 2857 1 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  {crab 3134  Vcvv 3469   ⊆ wss 3908  𝒫 cpw 4511   ↦ cmpt 5122  ‘cfv 6334  Basecbs 16474  occoc 16564  glbcglb 17544  LHypclh 37238  DVecHcdvh 38332  DIsoHcdih 38482  ocHcoch 38601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-doch 38602 This theorem is referenced by:  dochval2  38606  dochcl  38607  dochvalr  38611  dochss  38619
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