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Theorem docafvalN 36230
 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 36229 . . . 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 6180 . . 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, 7syl5eq 2666 . 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 6178 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
10 docaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
119, 10syl6eqr 2672 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
1211pweqd 4154 . . . 4 (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
13 fveq2 6178 . . . . . 6 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
14 docaval.i . . . . . 6 𝐼 = ((DIsoA‘𝐾)‘𝑊)
1513, 14syl6eqr 2672 . . . . 5 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐼)
1615cnveqd 5287 . . . . . . . . 9 (𝑤 = 𝑊((DIsoA‘𝐾)‘𝑤) = 𝐼)
1715rneqd 5342 . . . . . . . . . . 11 (𝑤 = 𝑊 → ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼)
18 rabeq 3187 . . . . . . . . . . 11 (ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑥𝑧})
1917, 18syl 17 . . . . . . . . . 10 (𝑤 = 𝑊 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑥𝑧})
2019inteqd 4471 . . . . . . . . 9 (𝑤 = 𝑊 {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧} = {𝑧 ∈ ran 𝐼𝑥𝑧})
2116, 20fveq12d 6184 . . . . . . . 8 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧}) = (𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧}))
2221fveq2d 6182 . . . . . . 7 (𝑤 = 𝑊 → ( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) = ( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})))
23 fveq2 6178 . . . . . . 7 (𝑤 = 𝑊 → ( 𝑤) = ( 𝑊))
2422, 23oveq12d 6653 . . . . . 6 (𝑤 = 𝑊 → (( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) = (( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)))
25 id 22 . . . . . 6 (𝑤 = 𝑊𝑤 = 𝑊)
2624, 25oveq12d 6653 . . . . 5 (𝑤 = 𝑊 → ((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤) = ((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))
2715, 26fveq12d 6184 . . . 4 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊)))
2812, 27mpteq12dv 4724 . . 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))))
29 eqid 2620 . . 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))))
30 fvex 6188 . . . . . 6 ((LTrn‘𝐾)‘𝑊) ∈ V
3110, 30eqeltri 2695 . . . . 5 𝑇 ∈ V
3231pwex 4839 . . . 4 𝒫 𝑇 ∈ V
3332mptex 6471 . . 3 (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))) ∈ V
3428, 29, 33fvmpt 6269 . 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))))
358, 34sylan9eq 2674 1 ((𝐾𝑉𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  {crab 2913  Vcvv 3195   ⊆ wss 3567  𝒫 cpw 4149  ∩ cint 4466   ↦ cmpt 4720  ◡ccnv 5103  ran crn 5105  ‘cfv 5876  (class class class)co 6635  occoc 15930  joincjn 16925  meetcmee 16926  LHypclh 35089  LTrncltrn 35206  DIsoAcdia 36136  ocAcocaN 36227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-docaN 36228 This theorem is referenced by:  docavalN  36231
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