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Theorem docaffvalN 40078
Description: Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
docaval.j ∨ = (joinβ€˜πΎ)
docaval.m ∧ = (meetβ€˜πΎ)
docaval.o βŠ₯ = (ocβ€˜πΎ)
docaval.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
docaffvalN (𝐾 ∈ 𝑉 β†’ (ocAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))))
Distinct variable groups:   𝑀,𝐻   π‘₯,𝑀,𝑧,𝐾
Allowed substitution hints:   𝐻(π‘₯,𝑧)   ∨ (π‘₯,𝑧,𝑀)   ∧ (π‘₯,𝑧,𝑀)   βŠ₯ (π‘₯,𝑧,𝑀)   𝑉(π‘₯,𝑧,𝑀)

Proof of Theorem docaffvalN
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6891 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 docaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2790 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6891 . . . . . . 7 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
65fveq1d 6893 . . . . . 6 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
76pweqd 4619 . . . . 5 (π‘˜ = 𝐾 β†’ 𝒫 ((LTrnβ€˜π‘˜)β€˜π‘€) = 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€))
8 fveq2 6891 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DIsoAβ€˜π‘˜) = (DIsoAβ€˜πΎ))
98fveq1d 6893 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DIsoAβ€˜π‘˜)β€˜π‘€) = ((DIsoAβ€˜πΎ)β€˜π‘€))
10 fveq2 6891 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = (meetβ€˜πΎ))
11 docaval.m . . . . . . . 8 ∧ = (meetβ€˜πΎ)
1210, 11eqtr4di 2790 . . . . . . 7 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = ∧ )
13 fveq2 6891 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
14 docaval.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
1513, 14eqtr4di 2790 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
16 fveq2 6891 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = (ocβ€˜πΎ))
17 docaval.o . . . . . . . . . 10 βŠ₯ = (ocβ€˜πΎ)
1816, 17eqtr4di 2790 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = βŠ₯ )
199cnveqd 5875 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ β—‘((DIsoAβ€˜π‘˜)β€˜π‘€) = β—‘((DIsoAβ€˜πΎ)β€˜π‘€))
209rneqd 5937 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) = ran ((DIsoAβ€˜πΎ)β€˜π‘€))
2120rabeqdv 3447 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧} = {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})
2221inteqd 4955 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ ∩ {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧} = ∩ {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})
2319, 22fveq12d 6898 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}) = (β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))
2418, 23fveq12d 6898 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) = ( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})))
2518fveq1d 6893 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((ocβ€˜π‘˜)β€˜π‘€) = ( βŠ₯ β€˜π‘€))
2615, 24, 25oveq123d 7432 . . . . . . 7 (π‘˜ = 𝐾 β†’ (((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))(joinβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘€)) = (( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)))
27 eqidd 2733 . . . . . . 7 (π‘˜ = 𝐾 β†’ 𝑀 = 𝑀)
2812, 26, 27oveq123d 7432 . . . . . 6 (π‘˜ = 𝐾 β†’ ((((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))(joinβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘€))(meetβ€˜π‘˜)𝑀) = ((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))
299, 28fveq12d 6898 . . . . 5 (π‘˜ = 𝐾 β†’ (((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜((((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))(joinβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘€))(meetβ€˜π‘˜)𝑀)) = (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))
307, 29mpteq12dv 5239 . . . 4 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ 𝒫 ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜((((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))(joinβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘€))(meetβ€˜π‘˜)𝑀))) = (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))))
314, 30mpteq12dv 5239 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜((((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))(joinβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘€))(meetβ€˜π‘˜)𝑀)))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))))
32 df-docaN 40077 . . 3 ocA = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜((((ocβ€˜π‘˜)β€˜(β—‘((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}))(joinβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘€))(meetβ€˜π‘˜)𝑀)))))
3331, 32, 3mptfvmpt 7232 . 2 (𝐾 ∈ V β†’ (ocAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))))
341, 33syl 17 1 (𝐾 ∈ 𝑉 β†’ (ocAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   βŠ† wss 3948  π’« cpw 4602  βˆ© cint 4950   ↦ cmpt 5231  β—‘ccnv 5675  ran crn 5677  β€˜cfv 6543  (class class class)co 7411  occoc 17207  joincjn 18266  meetcmee 18267  LHypclh 38941  LTrncltrn 39058  DIsoAcdia 39985  ocAcocaN 40076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-docaN 40077
This theorem is referenced by:  docafvalN  40079
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