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Theorem djafvalN 39535
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 39534 . . . 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 6841 . . 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 2789 . 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 6839 . . . . . 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 2795 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
98pweqd 4575 . . . 4 (𝑀 = π‘Š β†’ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) = 𝒫 𝑇)
10 fveq2 6839 . . . . . 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 2795 . . . . 5 (𝑀 = π‘Š β†’ ((ocAβ€˜πΎ)β€˜π‘€) = βŠ₯ )
1312fveq1d 6841 . . . . . 6 (𝑀 = π‘Š β†’ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) = ( βŠ₯ β€˜π‘₯))
1412fveq1d 6841 . . . . . 6 (𝑀 = π‘Š β†’ (((ocAβ€˜πΎ)β€˜π‘€)β€˜π‘¦) = ( βŠ₯ β€˜π‘¦))
1513, 14ineq12d 4171 . . . . 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 6846 . . . 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 7426 . . 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 2737 . . 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 6853 . . . . 5 𝑇 ∈ V
2019pwex 5333 . . . 4 𝒫 𝑇 ∈ V
2120, 20mpoex 8004 . . 3 (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))) ∈ V
2217, 18, 21fvmpt 6945 . 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 2797 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 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3907  π’« cpw 4558   ↦ cmpt 5186  β€˜cfv 6493   ∈ cmpo 7353  LHypclh 38385  LTrncltrn 38502  DIsoAcdia 39429  ocAcocaN 39520  vAcdjaN 39532
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-djaN 39533
This theorem is referenced by:  djavalN  39536
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