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Theorem docaffvalN 39584
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 3463 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6842 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 docaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2794 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6842 . . . . . . 7 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6844 . . . . . 6 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
76pweqd 4577 . . . . 5 (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8 fveq2 6842 . . . . . . 7 (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
98fveq1d 6844 . . . . . 6 (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
10 fveq2 6842 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
11 docaval.m . . . . . . . 8 = (meet‘𝐾)
1210, 11eqtr4di 2794 . . . . . . 7 (𝑘 = 𝐾 → (meet‘𝑘) = )
13 fveq2 6842 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 docaval.j . . . . . . . . 9 = (join‘𝐾)
1513, 14eqtr4di 2794 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = )
16 fveq2 6842 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 docaval.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17eqtr4di 2794 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
199cnveqd 5831 . . . . . . . . . 10 (𝑘 = 𝐾((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
209rneqd 5893 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ran ((DIsoA‘𝑘)‘𝑤) = ran ((DIsoA‘𝐾)‘𝑤))
2120rabeqdv 3422 . . . . . . . . . . 11 (𝑘 = 𝐾 → {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2221inteqd 4912 . . . . . . . . . 10 (𝑘 = 𝐾 {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})
2319, 22fveq12d 6849 . . . . . . . . 9 (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}) = (((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧}))
2418, 23fveq12d 6849 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧})) = ( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})))
2518fveq1d 6844 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ( 𝑤))
2615, 24, 25oveq123d 7378 . . . . . . 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 2737 . . . . . . 7 (𝑘 = 𝐾𝑤 = 𝑤)
2812, 26, 27oveq123d 7378 . . . . . 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 6849 . . . . 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 5196 . . . 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 5196 . . 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 39583 . . 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 7178 . 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 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560   cint 4907  cmpt 5188  ccnv 5632  ran crn 5634  cfv 6496  (class class class)co 7357  occoc 17141  joincjn 18200  meetcmee 18201  LHypclh 38447  LTrncltrn 38564  DIsoAcdia 39491  ocAcocaN 39582
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-docaN 39583
This theorem is referenced by:  docafvalN  39585
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