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Theorem djafvalN 38429
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 38428 . . . 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 6651 . . 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 2848 . 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 6649 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7 djaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
86, 7eqtr4di 2854 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
98pweqd 4519 . . . 4 (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
10 fveq2 6649 . . . . . 6 (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ((ocA‘𝐾)‘𝑊))
11 djaval.n . . . . . 6 = ((ocA‘𝐾)‘𝑊)
1210, 11eqtr4di 2854 . . . . 5 (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = )
1312fveq1d 6651 . . . . . 6 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑥) = ( 𝑥))
1412fveq1d 6651 . . . . . 6 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑦) = ( 𝑦))
1513, 14ineq12d 4143 . . . . 5 (𝑤 = 𝑊 → ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)) = (( 𝑥) ∩ ( 𝑦)))
1612, 15fveq12d 6656 . . . 4 (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))) = ( ‘(( 𝑥) ∩ ( 𝑦))))
179, 9, 16mpoeq123dv 7212 . . 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 2801 . . 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 6663 . . . . 5 𝑇 ∈ V
2019pwex 5249 . . . 4 𝒫 𝑇 ∈ V
2120, 20mpoex 7764 . . 3 (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) ∈ V
2217, 18, 21fvmpt 6749 . 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 2856 1 ((𝐾𝑉𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cin 3883  𝒫 cpw 4500  cmpt 5113  cfv 6328  cmpo 7141  LHypclh 37279  LTrncltrn 37396  DIsoAcdia 38323  ocAcocaN 38414  vAcdjaN 38426
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-djaN 38427
This theorem is referenced by:  djavalN  38430
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