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Theorem docaffvalN 39440
Description: Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
docaval.j = (join‘𝐾)
docaval.m = (meet‘𝐾)
docaval.o = (oc‘𝐾)
docaval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
docaffvalN (𝐾𝑉 → (ocA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))
Distinct variable groups:   𝑤,𝐻   𝑥,𝑤,𝑧,𝐾
Allowed substitution hints:   𝐻(𝑥,𝑧)   (𝑥,𝑧,𝑤)   (𝑥,𝑧,𝑤)   (𝑥,𝑧,𝑤)   𝑉(𝑥,𝑧,𝑤)

Proof of Theorem docaffvalN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3460 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6829 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 docaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2795 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6829 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6831 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
76pweqd 4568 . . . . 5 (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8 fveq2 6829 . . . . . . 7 (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
98fveq1d 6831 . . . . . 6 (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
10 fveq2 6829 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
11 docaval.m . . . . . . . 8 = (meet‘𝐾)
1210, 11eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (meet‘𝑘) = )
13 fveq2 6829 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 docaval.j . . . . . . . . 9 = (join‘𝐾)
1513, 14eqtr4di 2795 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = )
16 fveq2 6829 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 docaval.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17eqtr4di 2795 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
199cnveqd 5821 . . . . . . . . . 10 (𝑘 = 𝐾((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
209rneqd 5883 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ran ((DIsoA‘𝑘)‘𝑤) = ran ((DIsoA‘𝐾)‘𝑤))
2120rabeqdv 3419 . . . . . . . . . . 11 (𝑘 = 𝐾 → {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2221inteqd 4903 . . . . . . . . . 10 (𝑘 = 𝐾 {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2319, 22fveq12d 6836 . . . . . . . . 9 (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}) = (((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧}))
2418, 23fveq12d 6836 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧})) = ( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})))
2518fveq1d 6831 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ( 𝑤))
2615, 24, 25oveq123d 7362 . . . . . . 7 (𝑘 = 𝐾 → (((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤)) = (( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)))
27 eqidd 2738 . . . . . . 7 (𝑘 = 𝐾𝑤 = 𝑤)
2812, 26, 27oveq123d 7362 . . . . . 6 (𝑘 = 𝐾 → ((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤) = ((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))
299, 28fveq12d 6836 . . . . 5 (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)) = (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))
307, 29mpteq12dv 5187 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))) = (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))))
314, 30mpteq12dv 5187 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))
32 df-docaN 39439 . . 3 ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
3331, 32, 3mptfvmpt 7164 . 2 (𝐾 ∈ V → (ocA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))
341, 33syl 17 1 (𝐾𝑉 → (ocA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3404  Vcvv 3442  wss 3901  𝒫 cpw 4551   cint 4898  cmpt 5179  ccnv 5623  ran crn 5625  cfv 6483  (class class class)co 7341  occoc 17067  joincjn 18126  meetcmee 18127  LHypclh 38303  LTrncltrn 38420  DIsoAcdia 39347  ocAcocaN 39438
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 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pr 5376
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-ov 7344  df-docaN 39439
This theorem is referenced by:  docafvalN  39441
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