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Theorem djhfval 41380
Description: Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
Hypotheses
Ref Expression
djhval.h 𝐻 = (LHyp‘𝐾)
djhval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
djhval.v 𝑉 = (Base‘𝑈)
djhval.o = ((ocH‘𝐾)‘𝑊)
djhval.j = ((joinH‘𝐾)‘𝑊)
Assertion
Ref Expression
djhfval ((𝐾𝑋𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐻(𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem djhfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 djhval.j . . 3 = ((joinH‘𝐾)‘𝑊)
2 djhval.h . . . . 5 𝐻 = (LHyp‘𝐾)
32djhffval 41379 . . . 4 (𝐾𝑋 → (joinH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
43fveq1d 6909 . . 3 (𝐾𝑋 → ((joinH‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
51, 4eqtrid 2787 . 2 (𝐾𝑋 = ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
6 2fveq3 6912 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑊)))
7 djhval.v . . . . . . 7 𝑉 = (Base‘𝑈)
8 djhval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
98fveq2i 6910 . . . . . . 7 (Base‘𝑈) = (Base‘((DVecH‘𝐾)‘𝑊))
107, 9eqtri 2763 . . . . . 6 𝑉 = (Base‘((DVecH‘𝐾)‘𝑊))
116, 10eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
1211pweqd 4622 . . . 4 (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
13 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
14 djhval.o . . . . . 6 = ((ocH‘𝐾)‘𝑊)
1513, 14eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = )
1615fveq1d 6909 . . . . . 6 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑥) = ( 𝑥))
1715fveq1d 6909 . . . . . 6 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑦) = ( 𝑦))
1816, 17ineq12d 4229 . . . . 5 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)) = (( 𝑥) ∩ ( 𝑦)))
1915, 18fveq12d 6914 . . . 4 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))) = ( ‘(( 𝑥) ∩ ( 𝑦))))
2012, 12, 19mpoeq123dv 7508 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
21 eqid 2735 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))
227fvexi 6921 . . . . 5 𝑉 ∈ V
2322pwex 5386 . . . 4 𝒫 𝑉 ∈ V
2423, 23mpoex 8103 . . 3 (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) ∈ V
2520, 21, 24fvmpt 7016 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))‘𝑊) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
265, 25sylan9eq 2795 1 ((𝐾𝑋𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cin 3962  𝒫 cpw 4605  cmpt 5231  cfv 6563  cmpo 7433  Basecbs 17245  LHypclh 39967  DVecHcdvh 41061  ocHcoch 41330  joinHcdjh 41377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-djh 41378
This theorem is referenced by:  djhval  41381
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