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Theorem djafvalN 40544
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 40543 . . . 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 6893 . . 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, 4eqtrid 2779 . 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 6891 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7 djaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
86, 7eqtr4di 2785 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
98pweqd 4615 . . . 4 (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
10 fveq2 6891 . . . . . 6 (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ((ocA‘𝐾)‘𝑊))
11 djaval.n . . . . . 6 = ((ocA‘𝐾)‘𝑊)
1210, 11eqtr4di 2785 . . . . 5 (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = )
1312fveq1d 6893 . . . . . 6 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑥) = ( 𝑥))
1412fveq1d 6893 . . . . . 6 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑦) = ( 𝑦))
1513, 14ineq12d 4209 . . . . 5 (𝑤 = 𝑊 → ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)) = (( 𝑥) ∩ ( 𝑦)))
1612, 15fveq12d 6898 . . . 4 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))) = ( ‘(( 𝑥) ∩ ( 𝑦))))
179, 9, 16mpoeq123dv 7489 . . 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 2727 . . 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)))))
197fvexi 6905 . . . . 5 𝑇 ∈ V
2019pwex 5374 . . . 4 𝒫 𝑇 ∈ V
2120, 20mpoex 8078 . . 3 (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) ∈ V
2217, 18, 21fvmpt 6999 . 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)))))
235, 22sylan9eq 2787 1 ((𝐾𝑉𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cin 3943  𝒫 cpw 4598  cmpt 5225  cfv 6542  cmpo 7416  LHypclh 39394  LTrncltrn 39511  DIsoAcdia 40438  ocAcocaN 40529  vAcdjaN 40541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-djaN 40542
This theorem is referenced by:  djavalN  40545
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