Theorem djafvalN 42111

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 42110	. . . 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 6875	. . 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 2807	. 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 6873	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7		djaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 7	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
9	8	pweqd 4573	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
10		fveq2 6873	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ((ocA‘𝐾)‘𝑊))
11		djaval.n	. . . . . 6 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ((ocA‘𝐾)‘𝑤) = ⊥ )
13	12	fveq1d 6875	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
14	12	fveq1d 6875	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘𝑦) = ( ⊥ ‘𝑦))
15	13, 14	ineq12d 4166	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)) = (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))
16	12, 15	fveq12d 6880	. . . 4 ⊢ (𝑤 = 𝑊 → (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
17	9, 9, 16	mpoeq123dv 7483	. . 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 2760	. . 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 6887	. . . . 5 ⊢ 𝑇 ∈ V
20	19	pwex 5341	. . . 4 ⊢ 𝒫 𝑇 ∈ V
21	20, 20	mpoex 8075	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) ∈ V
22	17, 18, 21	fvmpt 6981	. 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 2815	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3897  𝒫 cpw 4556   ↦ cmpt 5185  ‘cfv 6527   ∈ cmpo 7410  LHypclh 40961  LTrncltrn 41078  DIsoAcdia 42005  ocAcocaN 42096  vAcdjaN 42108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-djaN 42109

This theorem is used by:  djavalN  42112