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Theorem docafvalN 41124
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)
docaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
docaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
docaval.n 𝑁 = ((ocA‘𝐾)‘𝑊)
Assertion
Ref Expression
docafvalN ((𝐾𝑉𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
Distinct variable groups:   𝑥,𝑧,𝐾   𝑥,𝐼,𝑧   𝑥,𝑇   𝑥,𝑊,𝑧
Allowed substitution hints:   𝑇(𝑧)   𝐻(𝑥,𝑧)   (𝑥,𝑧)   (𝑥,𝑧)   𝑁(𝑥,𝑧)   (𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem docafvalN
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 docaval.n . . 3 𝑁 = ((ocA‘𝐾)‘𝑊)
2 docaval.j . . . . 5 = (join‘𝐾)
3 docaval.m . . . . 5 = (meet‘𝐾)
4 docaval.o . . . . 5 = (oc‘𝐾)
5 docaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
62, 3, 4, 5docaffvalN 41123 . . . 4 (𝐾𝑉 → (ocA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))
76fveq1d 6908 . . 3 (𝐾𝑉 → ((ocA‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))))‘𝑊))
81, 7eqtrid 2789 . 2 (𝐾𝑉𝑁 = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))))‘𝑊))
9 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
10 docaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
119, 10eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
1211pweqd 4617 . . . 4 (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
13 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
14 docaval.i . . . . . 6 𝐼 = ((DIsoA‘𝐾)‘𝑊)
1513, 14eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐼)
1615cnveqd 5886 . . . . . . . . 9 (𝑤 = 𝑊((DIsoA‘𝐾)‘𝑤) = 𝐼)
1715rneqd 5949 . . . . . . . . . . 11 (𝑤 = 𝑊 → ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼)
1817rabeqdv 3452 . . . . . . . . . 10 (𝑤 = 𝑊 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑥𝑧})
1918inteqd 4951 . . . . . . . . 9 (𝑤 = 𝑊 {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑥𝑧})
2016, 19fveq12d 6913 . . . . . . . 8 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧}) = (𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧}))
2120fveq2d 6910 . . . . . . 7 (𝑤 = 𝑊 → ( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) = ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})))
22 fveq2 6906 . . . . . . 7 (𝑤 = 𝑊 → ( 𝑤) = ( 𝑊))
2321, 22oveq12d 7449 . . . . . 6 (𝑤 = 𝑊 → (( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) = (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)))
24 id 22 . . . . . 6 (𝑤 = 𝑊𝑤 = 𝑊)
2523, 24oveq12d 7449 . . . . 5 (𝑤 = 𝑊 → ((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤) = ((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))
2615, 25fveq12d 6913 . . . 4 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))
2712, 26mpteq12dv 5233 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
28 eqid 2737 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))))
2910fvexi 6920 . . . . 5 𝑇 ∈ V
3029pwex 5380 . . . 4 𝒫 𝑇 ∈ V
3130mptex 7243 . . 3 (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))) ∈ V
3227, 28, 31fvmpt 7016 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤))))‘𝑊) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
338, 32sylan9eq 2797 1 ((𝐾𝑉𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600   cint 4946  cmpt 5225  ccnv 5684  ran crn 5686  cfv 6561  (class class class)co 7431  occoc 17305  joincjn 18357  meetcmee 18358  LHypclh 39986  LTrncltrn 40103  DIsoAcdia 41030  ocAcocaN 41121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-docaN 41122
This theorem is referenced by:  docavalN  41125
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