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Theorem djafvalN 40005
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 = (LHypβ€˜πΎ)
djaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
djaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
djaval.n βŠ₯ = ((ocAβ€˜πΎ)β€˜π‘Š)
djaval.j 𝐽 = ((vAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
djafvalN ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐽 = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝑦,𝐾   π‘₯,𝑇,𝑦   π‘₯,π‘Š,𝑦
Allowed substitution hints:   𝐻(π‘₯,𝑦)   𝐼(π‘₯,𝑦)   𝐽(π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦)   𝑉(π‘₯,𝑦)

Proof of Theorem djafvalN
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 djaval.j . . 3 𝐽 = ((vAβ€˜πΎ)β€˜π‘Š)
2 djaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
32djaffvalN 40004 . . . 4 (𝐾 ∈ 𝑉 β†’ (vAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦))))))
43fveq1d 6894 . . 3 (𝐾 ∈ 𝑉 β†’ ((vAβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦)))))β€˜π‘Š))
51, 4eqtrid 2785 . 2 (𝐾 ∈ 𝑉 β†’ 𝐽 = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦)))))β€˜π‘Š))
6 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
7 djaval.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
86, 7eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
98pweqd 4620 . . . 4 (𝑀 = π‘Š β†’ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) = 𝒫 𝑇)
10 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ ((ocAβ€˜πΎ)β€˜π‘€) = ((ocAβ€˜πΎ)β€˜π‘Š))
11 djaval.n . . . . . 6 βŠ₯ = ((ocAβ€˜πΎ)β€˜π‘Š)
1210, 11eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ ((ocAβ€˜πΎ)β€˜π‘€) = βŠ₯ )
1312fveq1d 6894 . . . . . 6 (𝑀 = π‘Š β†’ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) = ( βŠ₯ β€˜π‘₯))
1412fveq1d 6894 . . . . . 6 (𝑀 = π‘Š β†’ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦) = ( βŠ₯ β€˜π‘¦))
1513, 14ineq12d 4214 . . . . 5 (𝑀 = π‘Š β†’ ((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦)) = (( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))
1612, 15fveq12d 6899 . . . 4 (𝑀 = π‘Š β†’ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦))) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))
179, 9, 16mpoeq123dv 7484 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦)))) = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
18 eqid 2733 . . 3 (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦))))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦)))))
197fvexi 6906 . . . . 5 𝑇 ∈ V
2019pwex 5379 . . . 4 𝒫 𝑇 ∈ V
2120, 20mpoex 8066 . . 3 (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))) ∈ V
2217, 18, 21fvmpt 6999 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€), 𝑦 ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((ocAβ€˜πΎ)β€˜π‘€)β€˜((((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) ∩ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦)))))β€˜π‘Š) = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
235, 22sylan9eq 2793 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐽 = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∩ cin 3948  π’« cpw 4603   ↦ cmpt 5232  β€˜cfv 6544   ∈ cmpo 7411  LHypclh 38855  LTrncltrn 38972  DIsoAcdia 39899  ocAcocaN 39990  vAcdjaN 40002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-djaN 40003
This theorem is referenced by:  djavalN  40006
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