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Theorem dochfval 37153
 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
dochfval ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
Distinct variable groups:   𝑦,𝐵   𝑥,𝑦,𝐾   𝑥,𝑉   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝑈(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝑁(𝑥,𝑦)   (𝑥,𝑦)   𝑉(𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem dochfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dochval.n . . 3 𝑁 = ((ocH‘𝐾)‘𝑊)
2 dochval.b . . . . 5 𝐵 = (Base‘𝐾)
3 dochval.g . . . . 5 𝐺 = (glb‘𝐾)
4 dochval.o . . . . 5 = (oc‘𝐾)
5 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
62, 3, 4, 5dochffval 37152 . . . 4 (𝐾𝑋 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
76fveq1d 6334 . . 3 (𝐾𝑋 → ((ocH‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
81, 7syl5eq 2816 . 2 (𝐾𝑋𝑁 = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
9 fveq2 6332 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
10 dochval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
119, 10syl6eqr 2822 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1211fveq2d 6336 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
13 dochval.v . . . . . 6 𝑉 = (Base‘𝑈)
1412, 13syl6eqr 2822 . . . . 5 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
1514pweqd 4300 . . . 4 (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
16 fveq2 6332 . . . . . 6 (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = ((DIsoH‘𝐾)‘𝑊))
17 dochval.i . . . . . 6 𝐼 = ((DIsoH‘𝐾)‘𝑊)
1816, 17syl6eqr 2822 . . . . 5 (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = 𝐼)
1918fveq1d 6334 . . . . . . . . 9 (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘𝑦) = (𝐼𝑦))
2019sseq2d 3780 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (𝐼𝑦)))
2120rabbidv 3338 . . . . . . 7 (𝑤 = 𝑊 → {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (𝐼𝑦)})
2221fveq2d 6336 . . . . . 6 (𝑤 = 𝑊 → (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))
2322fveq2d 6336 . . . . 5 (𝑤 = 𝑊 → ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))
2418, 23fveq12d 6338 . . . 4 (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)}))))
2515, 24mpteq12dv 4865 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
26 eqid 2770 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))
27 fvex 6342 . . . . . 6 (Base‘𝑈) ∈ V
2813, 27eqeltri 2845 . . . . 5 𝑉 ∈ V
2928pwex 4976 . . . 4 𝒫 𝑉 ∈ V
3029mptex 6629 . . 3 (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))) ∈ V
3125, 26, 30fvmpt 6424 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
328, 31sylan9eq 2824 1 ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  {crab 3064  Vcvv 3349   ⊆ wss 3721  𝒫 cpw 4295   ↦ cmpt 4861  ‘cfv 6031  Basecbs 16063  occoc 16156  glbcglb 17150  LHypclh 35785  DVecHcdvh 36881  DIsoHcdih 37031  ocHcoch 37150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-doch 37151 This theorem is referenced by:  dochval  37154  dochfN  37159
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