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Theorem docaffvalN 39587
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 3464 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6843 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 docaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2795 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6843 . . . . . . 7 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
65fveq1d 6845 . . . . . 6 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
76pweqd 4578 . . . . 5 (π‘˜ = 𝐾 β†’ 𝒫 ((LTrnβ€˜π‘˜)β€˜π‘€) = 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€))
8 fveq2 6843 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DIsoAβ€˜π‘˜) = (DIsoAβ€˜πΎ))
98fveq1d 6845 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DIsoAβ€˜π‘˜)β€˜π‘€) = ((DIsoAβ€˜πΎ)β€˜π‘€))
10 fveq2 6843 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = (meetβ€˜πΎ))
11 docaval.m . . . . . . . 8 ∧ = (meetβ€˜πΎ)
1210, 11eqtr4di 2795 . . . . . . 7 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = ∧ )
13 fveq2 6843 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
14 docaval.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
1513, 14eqtr4di 2795 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
16 fveq2 6843 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = (ocβ€˜πΎ))
17 docaval.o . . . . . . . . . 10 βŠ₯ = (ocβ€˜πΎ)
1816, 17eqtr4di 2795 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = βŠ₯ )
199cnveqd 5832 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ β—‘((DIsoAβ€˜π‘˜)β€˜π‘€) = β—‘((DIsoAβ€˜πΎ)β€˜π‘€))
209rneqd 5894 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) = ran ((DIsoAβ€˜πΎ)β€˜π‘€))
2120rabeqdv 3423 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ {𝑧 ∈ ran ((DIsoAβ€˜π‘˜)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧} = {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})
2221inteqd 4913 . . . . . . . . . 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 6850 . . . . . . . . 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 6850 . . . . . . . 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 6845 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((ocβ€˜π‘˜)β€˜π‘€) = ( βŠ₯ β€˜π‘€))
2615, 24, 25oveq123d 7379 . . . . . . 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 2738 . . . . . . 7 (π‘˜ = 𝐾 β†’ 𝑀 = 𝑀)
2812, 26, 27oveq123d 7379 . . . . . 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 6850 . . . . 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 5197 . . . 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 5197 . . 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 39586 . . 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 7179 . 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 1542   ∈ wcel 2107  {crab 3408  Vcvv 3446   βŠ† wss 3911  π’« cpw 4561  βˆ© cint 4908   ↦ cmpt 5189  β—‘ccnv 5633  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  occoc 17142  joincjn 18201  meetcmee 18202  LHypclh 38450  LTrncltrn 38567  DIsoAcdia 39494  ocAcocaN 39585
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-docaN 39586
This theorem is referenced by:  docafvalN  39588
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