Theorem djafvalN 41136

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 = (LHyp‘𝐾)
djaval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
djaval.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
djaval.n	⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
djaval.j	⊢ 𝐽 = ((vA‘𝐾)‘𝑊)

Assertion

Ref	Expression
djafvalN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝑇,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝐽(𝑥,𝑦)   ⊥ (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem djafvalN

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djaval.j	. . 3 ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)
2		djaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	djaffvalN 41135	. . . 4 ⊢ (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))))))
4	3	fveq1d 6908	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((vA‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
5	1, 4	eqtrid 2789	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐽 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
6		fveq2 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7		djaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 7	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
9	8	pweqd 4617	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
10		fveq2 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ((ocA‘𝐾)‘𝑊))
11		djaval.n	. . . . . 6 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ⊥ )
13	12	fveq1d 6908	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
14	12	fveq1d 6908	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑦) = ( ⊥ ‘𝑦))
15	13, 14	ineq12d 4221	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)) = (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))
16	12, 15	fveq12d 6913	. . . 4 ⊢ (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
17	9, 9, 16	mpoeq123dv 7508	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
18		eqid 2737	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (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)))))
19	7	fvexi 6920	. . . . 5 ⊢ 𝑇 ∈ V
20	19	pwex 5380	. . . 4 ⊢ 𝒫 𝑇 ∈ V
21	20, 20	mpoex 8104	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) ∈ V
22	17, 18, 21	fvmpt 7016	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))‘𝑊) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
23	5, 22	sylan9eq 2797	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∩ cin 3950  𝒫 cpw 4600   ↦ cmpt 5225  ‘cfv 6561   ∈ cmpo 7433  LHypclh 39986  LTrncltrn 40103  DIsoAcdia 41030  ocAcocaN 41121  vAcdjaN 41133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-djaN 41134

This theorem is referenced by:  djavalN  41137