Theorem djaffvalN 39074

Description: Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
djaval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
djaffvalN	⊢ (𝐾 ∈ 𝑉 → (vA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))

Distinct variable groups:   𝑤,𝐻   𝑥,𝑤,𝑦,𝐾

Allowed substitution hints:   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑤)

Proof of Theorem djaffvalN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6756	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		djaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2797	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6758	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7	6	pweqd 4549	. . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (ocA‘𝑘) = (ocA‘𝐾))
9	8	fveq1d 6758	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((ocA‘𝑘)‘𝑤) = ((ocA‘𝐾)‘𝑤))
10	9	fveq1d 6758	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘𝑥) = (((ocA‘𝐾)‘𝑤)‘𝑥))
11	9	fveq1d 6758	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘𝑦) = (((ocA‘𝐾)‘𝑤)‘𝑦))
12	10, 11	ineq12d 4144	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦)) = ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))
13	9, 12	fveq12d 6763	. . . . 5 ⊢ (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))) = (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))
14	7, 7, 13	mpoeq123dv 7328	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))
15	4, 14	mpteq12dv 5161	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
16		df-djaN 39073	. . 3 ⊢ vA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))))
17	15, 16, 3	mptfvmpt 7086	. 2 ⊢ (𝐾 ∈ V → (vA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
18	1, 17	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (vA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  𝒫 cpw 4530   ↦ cmpt 5153  ‘cfv 6418   ∈ cmpo 7257  LHypclh 37925  LTrncltrn 38042  ocAcocaN 39060  vAcdjaN 39072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-oprab 7259  df-mpo 7260  df-djaN 39073

This theorem is referenced by:  djafvalN  39075