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Theorem dochffval 35455
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)
Assertion
Ref Expression
dochffval (𝐾𝑉 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
Distinct variable groups:   𝑦,𝐵   𝑤,𝐻   𝑥,𝑤,𝑦,𝐾
Allowed substitution hints:   𝐵(𝑥,𝑤)   𝐺(𝑥,𝑦,𝑤)   𝐻(𝑥,𝑦)   (𝑥,𝑦,𝑤)   𝑉(𝑥,𝑦,𝑤)

Proof of Theorem dochffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3180 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6084 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dochval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2657 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6084 . . . . . . . 8 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
65fveq1d 6086 . . . . . . 7 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
76fveq2d 6088 . . . . . 6 (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
87pweqd 4108 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) = 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)))
9 fveq2 6084 . . . . . . 7 (𝑘 = 𝐾 → (DIsoH‘𝑘) = (DIsoH‘𝐾))
109fveq1d 6086 . . . . . 6 (𝑘 = 𝐾 → ((DIsoH‘𝑘)‘𝑤) = ((DIsoH‘𝐾)‘𝑤))
11 fveq2 6084 . . . . . . . 8 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
12 dochval.o . . . . . . . 8 = (oc‘𝐾)
1311, 12syl6eqr 2657 . . . . . . 7 (𝑘 = 𝐾 → (oc‘𝑘) = )
14 fveq2 6084 . . . . . . . . 9 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
15 dochval.g . . . . . . . . 9 𝐺 = (glb‘𝐾)
1614, 15syl6eqr 2657 . . . . . . . 8 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
17 fveq2 6084 . . . . . . . . . 10 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
18 dochval.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1917, 18syl6eqr 2657 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
2010fveq1d 6086 . . . . . . . . . 10 (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘𝑦) = (((DIsoH‘𝐾)‘𝑤)‘𝑦))
2120sseq2d 3591 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)))
2219, 21rabeqbidv 3163 . . . . . . . 8 (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)} = {𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})
2316, 22fveq12d 6090 . . . . . . 7 (𝑘 = 𝐾 → ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))
2413, 23fveq12d 6090 . . . . . 6 (𝑘 = 𝐾 → ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})) = ( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))
2510, 24fveq12d 6090 . . . . 5 (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))) = (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))
268, 25mpteq12dv 4653 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))) = (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))
274, 26mpteq12dv 4653 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
28 df-doch 35454 . . 3 ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))
29 fvex 6094 . . . . 5 (LHyp‘𝐾) ∈ V
303, 29eqeltri 2679 . . . 4 𝐻 ∈ V
3130mptex 6364 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))) ∈ V
3227, 28, 31fvmpt 6172 . 2 (𝐾 ∈ V → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
331, 32syl 17 1 (𝐾𝑉 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  wss 3535  𝒫 cpw 4103  cmpt 4633  cfv 5786  Basecbs 15637  occoc 15718  glbcglb 16708  LHypclh 34087  DVecHcdvh 35184  DIsoHcdih 35334  ocHcoch 35453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-doch 35454
This theorem is referenced by:  dochfval  35456
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