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

Theorem dochval 41353
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 41352 . . . 4 ((𝐾𝑌𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
109adantr 480 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
1110fveq1d 6908 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋))
127fvexi 6920 . . . . . 6 𝑉 ∈ V
1312elpw2 5334 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1413biimpri 228 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1514adantl 481 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑋 ∈ 𝒫 𝑉)
16 fvex 6919 . . 3 (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V
17 sseq1 4009 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼𝑦) ↔ 𝑋 ⊆ (𝐼𝑦)))
1817rabbidv 3444 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝐵𝑥 ⊆ (𝐼𝑦)} = {𝑦𝐵𝑋 ⊆ (𝐼𝑦)})
1918fveq2d 6910 . . . . . 6 (𝑥 = 𝑋 → (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}) = (𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))
2019fveq2d 6910 . . . . 5 (𝑥 = 𝑋 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)})))
2120fveq2d 6910 . . . 4 (𝑥 = 𝑋 → (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
22 eqid 2737 . . . 4 (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))
2321, 22fvmptg 7014 . . 3 ((𝑋 ∈ 𝒫 𝑉 ∧ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
2415, 16, 23sylancl 586 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
2511, 24eqtrd 2777 1 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600  cmpt 5225  cfv 6561  Basecbs 17247  occoc 17305  glbcglb 18356  LHypclh 39986  DVecHcdvh 41080  DIsoHcdih 41230  ocHcoch 41349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-doch 41350
This theorem is referenced by:  dochval2  41354  dochcl  41355  dochvalr  41359  dochss  41367
  Copyright terms: Public domain W3C validator