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Theorem dochffval 39862
Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
Hypotheses
Ref Expression
dochval.b 𝐡 = (Baseβ€˜πΎ)
dochval.g 𝐺 = (glbβ€˜πΎ)
dochval.o βŠ₯ = (ocβ€˜πΎ)
dochval.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
dochffval (𝐾 ∈ 𝑉 β†’ (ocHβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))))
Distinct variable groups:   𝑦,𝐡   𝑀,𝐻   π‘₯,𝑀,𝑦,𝐾
Allowed substitution hints:   𝐡(π‘₯,𝑀)   𝐺(π‘₯,𝑦,𝑀)   𝐻(π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦,𝑀)   𝑉(π‘₯,𝑦,𝑀)

Proof of Theorem dochffval
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3465 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6846 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 dochval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2791 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6846 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (DVecHβ€˜π‘˜) = (DVecHβ€˜πΎ))
65fveq1d 6848 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((DVecHβ€˜π‘˜)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘€))
76fveq2d 6850 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))
87pweqd 4581 . . . . 5 (π‘˜ = 𝐾 β†’ 𝒫 (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))
9 fveq2 6846 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DIsoHβ€˜π‘˜) = (DIsoHβ€˜πΎ))
109fveq1d 6848 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DIsoHβ€˜π‘˜)β€˜π‘€) = ((DIsoHβ€˜πΎ)β€˜π‘€))
11 fveq2 6846 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = (ocβ€˜πΎ))
12 dochval.o . . . . . . . 8 βŠ₯ = (ocβ€˜πΎ)
1311, 12eqtr4di 2791 . . . . . . 7 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = βŠ₯ )
14 fveq2 6846 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (glbβ€˜π‘˜) = (glbβ€˜πΎ))
15 dochval.g . . . . . . . . 9 𝐺 = (glbβ€˜πΎ)
1614, 15eqtr4di 2791 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (glbβ€˜π‘˜) = 𝐺)
17 fveq2 6846 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
18 dochval.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
1917, 18eqtr4di 2791 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
2010fveq1d 6848 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦) = (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦))
2120sseq2d 3980 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦) ↔ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)))
2219, 21rabeqbidv 3423 . . . . . . . 8 (π‘˜ = 𝐾 β†’ {𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)} = {𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})
2316, 22fveq12d 6853 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((glbβ€˜π‘˜)β€˜{𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)}) = (πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))
2413, 23fveq12d 6853 . . . . . 6 (π‘˜ = 𝐾 β†’ ((ocβ€˜π‘˜)β€˜((glbβ€˜π‘˜)β€˜{𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)})) = ( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))
2510, 24fveq12d 6853 . . . . 5 (π‘˜ = 𝐾 β†’ (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜((ocβ€˜π‘˜)β€˜((glbβ€˜π‘˜)β€˜{𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)}))) = (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))
268, 25mpteq12dv 5200 . . . 4 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜((ocβ€˜π‘˜)β€˜((glbβ€˜π‘˜)β€˜{𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)})))) = (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))))
274, 26mpteq12dv 5200 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜((ocβ€˜π‘˜)β€˜((glbβ€˜π‘˜)β€˜{𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)}))))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))))
28 df-doch 39861 . . 3 ocH = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜((ocβ€˜π‘˜)β€˜((glbβ€˜π‘˜)β€˜{𝑦 ∈ (Baseβ€˜π‘˜) ∣ π‘₯ βŠ† (((DIsoHβ€˜π‘˜)β€˜π‘€)β€˜π‘¦)}))))))
2927, 28, 3mptfvmpt 7182 . 2 (𝐾 ∈ V β†’ (ocHβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))))
301, 29syl 17 1 (𝐾 ∈ 𝑉 β†’ (ocHβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3447   βŠ† wss 3914  π’« cpw 4564   ↦ cmpt 5192  β€˜cfv 6500  Basecbs 17091  occoc 17149  glbcglb 18207  LHypclh 38497  DVecHcdvh 39591  DIsoHcdih 39741  ocHcoch 39860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-doch 39861
This theorem is referenced by:  dochfval  39863
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