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Theorem djafvalN 35441
 Description: Subspace join for DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djaval.h 𝐻 = (LHyp‘𝐾)
djaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
djaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
djaval.n = ((ocA‘𝐾)‘𝑊)
djaval.j 𝐽 = ((vA‘𝐾)‘𝑊)
Assertion
Ref Expression
djafvalN ((𝐾𝑉𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝑇,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝐽(𝑥,𝑦)   (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem djafvalN
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 djaval.j . . 3 𝐽 = ((vA‘𝐾)‘𝑊)
2 djaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
32djaffvalN 35440 . . . 4 (𝐾𝑉 → (vA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
43fveq1d 6105 . . 3 (𝐾𝑉 → ((vA‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
51, 4syl5eq 2656 . 2 (𝐾𝑉𝐽 = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
6 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7 djaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
86, 7syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
98pweqd 4113 . . . 4 (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
10 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ((ocA‘𝐾)‘𝑊))
11 djaval.n . . . . . 6 = ((ocA‘𝐾)‘𝑊)
1210, 11syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = )
1312fveq1d 6105 . . . . . 6 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑥) = ( 𝑥))
1412fveq1d 6105 . . . . . 6 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑦) = ( 𝑦))
1513, 14ineq12d 3777 . . . . 5 (𝑤 = 𝑊 → ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)) = (( 𝑥) ∩ ( 𝑦)))
1612, 15fveq12d 6109 . . . 4 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))) = ( ‘(( 𝑥) ∩ ( 𝑦))))
179, 9, 16mpt2eq123dv 6615 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
18 eqid 2610 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))
19 fvex 6113 . . . . . 6 ((LTrn‘𝐾)‘𝑊) ∈ V
207, 19eqeltri 2684 . . . . 5 𝑇 ∈ V
2120pwex 4774 . . . 4 𝒫 𝑇 ∈ V
2221, 21mpt2ex 7136 . . 3 (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) ∈ V
2317, 18, 22fvmpt 6191 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))‘𝑊) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
245, 23sylan9eq 2664 1 ((𝐾𝑉𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  𝒫 cpw 4108   ↦ cmpt 4643  ‘cfv 5804   ↦ cmpt2 6551  LHypclh 34288  LTrncltrn 34405  DIsoAcdia 35335  ocAcocaN 35426  vAcdjaN 35438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-djaN 35439 This theorem is referenced by:  djavalN  35442
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