Theorem djaffvalN 37821

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 3458	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6545	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		djaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	syl6eqr 2851	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6545	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6547	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7	6	pweqd 4464	. . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8		fveq2 6545	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (ocA‘𝑘) = (ocA‘𝐾))
9	8	fveq1d 6547	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((ocA‘𝑘)‘𝑤) = ((ocA‘𝐾)‘𝑤))
10	9	fveq1d 6547	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘𝑥) = (((ocA‘𝐾)‘𝑤)‘𝑥))
11	9	fveq1d 6547	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((ocA‘𝑘)‘𝑤)‘𝑦) = (((ocA‘𝐾)‘𝑤)‘𝑦))
12	10, 11	ineq12d 4116	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦)) = ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))
13	9, 12	fveq12d 6552	. . . . 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 7094	. . . 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 5052	. . 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 37820	. . 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 6863	. 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 1525   ∈ wcel 2083  Vcvv 3440   ∩ cin 3864  𝒫 cpw 4459   ↦ cmpt 5047  ‘cfv 6232   ∈ cmpo 7025  LHypclh 36672  LTrncltrn 36789  ocAcocaN 37807  vAcdjaN 37819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-oprab 7027  df-mpo 7028  df-djaN 37820

This theorem is referenced by:  djafvalN  37822