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Theorem djhfval 35704
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 35703 . . . 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 6105 . . 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, 4syl5eq 2656 . 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 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
76fveq2d 6107 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑊)))
8 djhval.v . . . . . . 7 𝑉 = (Base‘𝑈)
9 djhval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
109fveq2i 6106 . . . . . . 7 (Base‘𝑈) = (Base‘((DVecH‘𝐾)‘𝑊))
118, 10eqtri 2632 . . . . . 6 𝑉 = (Base‘((DVecH‘𝐾)‘𝑊))
127, 11syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
1312pweqd 4113 . . . 4 (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
14 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
15 djhval.o . . . . . 6 = ((ocH‘𝐾)‘𝑊)
1614, 15syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = )
1716fveq1d 6105 . . . . . 6 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑥) = ( 𝑥))
1816fveq1d 6105 . . . . . 6 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑦) = ( 𝑦))
1917, 18ineq12d 3777 . . . . 5 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)) = (( 𝑥) ∩ ( 𝑦)))
2016, 19fveq12d 6109 . . . 4 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))) = ( ‘(( 𝑥) ∩ ( 𝑦))))
2113, 13, 20mpt2eq123dv 6615 . . 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)))))
22 eqid 2610 . . 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)))))
23 fvex 6113 . . . . . 6 (Base‘𝑈) ∈ V
248, 23eqeltri 2684 . . . . 5 𝑉 ∈ V
2524pwex 4774 . . . 4 𝒫 𝑉 ∈ V
2625, 25mpt2ex 7136 . . 3 (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) ∈ V
2721, 22, 26fvmpt 6191 . 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)))))
285, 27sylan9eq 2664 1 ((𝐾𝑋𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 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  Basecbs 15695  LHypclh 34288  DVecHcdvh 35385  ocHcoch 35654  joinHcdjh 35701
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-djh 35702
This theorem is referenced by:  djhval  35705
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