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Theorem docaffvalN 36727
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 3243 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6229 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 docaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2703 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6229 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6231 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
76pweqd 4196 . . . . 5 (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8 fveq2 6229 . . . . . . 7 (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
98fveq1d 6231 . . . . . 6 (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
10 fveq2 6229 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
11 docaval.m . . . . . . . 8 = (meet‘𝐾)
1210, 11syl6eqr 2703 . . . . . . 7 (𝑘 = 𝐾 → (meet‘𝑘) = )
13 fveq2 6229 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 docaval.j . . . . . . . . 9 = (join‘𝐾)
1513, 14syl6eqr 2703 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = )
16 fveq2 6229 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 docaval.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17syl6eqr 2703 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
199cnveqd 5330 . . . . . . . . . 10 (𝑘 = 𝐾((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
209rneqd 5385 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ran ((DIsoA‘𝑘)‘𝑤) = ran ((DIsoA‘𝐾)‘𝑤))
21 rabeq 3223 . . . . . . . . . . . 12 (ran ((DIsoA‘𝑘)‘𝑤) = ran ((DIsoA‘𝐾)‘𝑤) → {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2220, 21syl 17 . . . . . . . . . . 11 (𝑘 = 𝐾 → {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2322inteqd 4512 . . . . . . . . . 10 (𝑘 = 𝐾 {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2419, 23fveq12d 6235 . . . . . . . . 9 (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}) = (((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧}))
2518, 24fveq12d 6235 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧})) = ( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})))
2618fveq1d 6231 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ( 𝑤))
2715, 25, 26oveq123d 6711 . . . . . . 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)))
28 eqidd 2652 . . . . . . 7 (𝑘 = 𝐾𝑤 = 𝑤)
2912, 27, 28oveq123d 6711 . . . . . 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))
309, 29fveq12d 6235 . . . . 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)))
317, 30mpteq12dv 4766 . . . 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))))
324, 31mpteq12dv 4766 . . 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)))))
33 df-docaN 36726 . . 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)))))
34 fvex 6239 . . . . 5 (LHyp‘𝐾) ∈ V
353, 34eqeltri 2726 . . . 4 𝐻 ∈ V
3635mptex 6527 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))) ∈ V
3732, 33, 36fvmpt 6321 . 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)))))
381, 37syl 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 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   cint 4507  cmpt 4762  ccnv 5142  ran crn 5144  cfv 5926  (class class class)co 6690  occoc 15996  joincjn 16991  meetcmee 16992  LHypclh 35588  LTrncltrn 35705  DIsoAcdia 36634  ocAcocaN 36725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-docaN 36726
This theorem is referenced by:  docafvalN  36728
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