Theorem djhval 41381

Description: Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)

Hypotheses

Ref	Expression
djhval.h	⊢ 𝐻 = (LHyp‘𝐾)
djhval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
djhval.v	⊢ 𝑉 = (Base‘𝑈)
djhval.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
djhval.j	⊢ ∨ = ((joinH‘𝐾)‘𝑊)

Assertion

Ref	Expression
djhval	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))

Proof of Theorem djhval

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

Step	Hyp	Ref	Expression
1		djhval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		djhval.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		djhval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
4		djhval.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
5		djhval.j	. . . . 5 ⊢ ∨ = ((joinH‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	djhfval 41380	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
7	6	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
8	7	oveqd 7448	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌))
9	3	fvexi 6921	. . . . . 6 ⊢ 𝑉 ∈ V
10	9	elpw2 5340	. . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉)
11	10	biimpri 228	. . . 4 ⊢ (𝑋 ⊆ 𝑉 → 𝑋 ∈ 𝒫 𝑉)
12	11	ad2antrl 728	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → 𝑋 ∈ 𝒫 𝑉)
13	9	elpw2 5340	. . . . 5 ⊢ (𝑌 ∈ 𝒫 𝑉 ↔ 𝑌 ⊆ 𝑉)
14	13	biimpri 228	. . . 4 ⊢ (𝑌 ⊆ 𝑉 → 𝑌 ∈ 𝒫 𝑉)
15	14	ad2antll 729	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → 𝑌 ∈ 𝒫 𝑉)
16		fvexd 6922	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∈ V)
17		fveq2 6907	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
18	17	ineq1d 4227	. . . . 5 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)))
19	18	fveq2d 6911	. . . 4 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))))
20		fveq2 6907	. . . . . 6 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
21	20	ineq2d 4228	. . . . 5 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))
22	21	fveq2d 6911	. . . 4 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
23		eqid 2735	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
24	19, 22, 23	ovmpog 7592	. . 3 ⊢ ((𝑋 ∈ 𝒫 𝑉 ∧ 𝑌 ∈ 𝒫 𝑉 ∧ ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∈ V) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
25	12, 15, 16, 24	syl3anc 1370	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
26	8, 25	eqtrd 2775	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  HLchlt 39332  LHypclh 39967  DVecHcdvh 41061  ocHcoch 41330  joinHcdjh 41377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-djh 41378

This theorem is referenced by:  djhval2  41382  djhcl  41383  djhlj  41384  djhexmid  41394