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Theorem dochval 41334
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 41333 . . . 4 ((𝐾𝑌𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
109adantr 480 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
1110fveq1d 6909 . 2 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))‘𝑋))
127fvexi 6921 . . . . . 6 𝑉 ∈ V
1312elpw2 5340 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1413biimpri 228 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1514adantl 481 . . 3 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → 𝑋 ∈ 𝒫 𝑉)
16 fvex 6920 . . 3 (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))) ∈ V
17 sseq1 4021 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼𝑦) ↔ 𝑋 ⊆ (𝐼𝑦)))
1817rabbidv 3441 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝐵𝑥 ⊆ (𝐼𝑦)} = {𝑦𝐵𝑋 ⊆ (𝐼𝑦)})
1918fveq2d 6911 . . . . . 6 (𝑥 = 𝑋 → (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}) = (𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))
2019fveq2d 6911 . . . . 5 (𝑥 = 𝑋 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)})))
2120fveq2d 6911 . . . 4 (𝑥 = 𝑋 → (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
22 eqid 2735 . . . 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 2775 1 (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605  cmpt 5231  cfv 6563  Basecbs 17245  occoc 17306  glbcglb 18368  LHypclh 39967  DVecHcdvh 41061  DIsoHcdih 41211  ocHcoch 41330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-doch 41331
This theorem is referenced by:  dochval2  41335  dochcl  41336  dochvalr  41340  dochss  41348
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