Theorem djavalN 41244

Description: Subspace join for DVecA partial vector space. (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
djavalN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))

Proof of Theorem djavalN

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		djaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		djaval.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		djaval.i	. . . . 5 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
4		djaval.n	. . . . 5 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
5		djaval.j	. . . . 5 ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	djafvalN 41243	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
7	6	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
8	7	oveqd 7363	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌))
9	2	fvexi 6836	. . . . . 6 ⊢ 𝑇 ∈ V
10	9	elpw2 5270	. . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑇 ↔ 𝑋 ⊆ 𝑇)
11	10	biimpri 228	. . . 4 ⊢ (𝑋 ⊆ 𝑇 → 𝑋 ∈ 𝒫 𝑇)
12	11	ad2antrl 728	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → 𝑋 ∈ 𝒫 𝑇)
13	9	elpw2 5270	. . . . 5 ⊢ (𝑌 ∈ 𝒫 𝑇 ↔ 𝑌 ⊆ 𝑇)
14	13	biimpri 228	. . . 4 ⊢ (𝑌 ⊆ 𝑇 → 𝑌 ∈ 𝒫 𝑇)
15	14	ad2antll 729	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → 𝑌 ∈ 𝒫 𝑇)
16		fvexd 6837	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∈ V)
17		fveq2 6822	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
18	17	ineq1d 4166	. . . . 5 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)))
19	18	fveq2d 6826	. . . 4 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))))
20		fveq2 6822	. . . . . 6 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
21	20	ineq2d 4167	. . . . 5 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))
22	21	fveq2d 6826	. . . 4 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
23		eqid 2731	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
24	19, 22, 23	ovmpog 7505	. . 3 ⊢ ((𝑋 ∈ 𝒫 𝑇 ∧ 𝑌 ∈ 𝒫 𝑇 ∧ ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∈ V) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
25	12, 15, 16, 24	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
26	8, 25	eqtrd 2766	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  HLchlt 39459  LHypclh 40093  LTrncltrn 40210  DIsoAcdia 41137  ocAcocaN 41228  vAcdjaN 41240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-djaN 41241

This theorem is referenced by:  djaclN  41245  djajN  41246