Theorem djhval 41890

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 41889	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
7	6	adantr 481	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
8	7	oveqd 7373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌))
9	3	fvexi 6841	. . . . . 6 ⊢ 𝑉 ∈ V
10	9	elpw2 5262	. . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉)
11	10	biimpri 229	. . . 4 ⊢ (𝑋 ⊆ 𝑉 → 𝑋 ∈ 𝒫 𝑉)
12	11	ad2antrl 734	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → 𝑋 ∈ 𝒫 𝑉)
13	9	elpw2 5262	. . . . 5 ⊢ (𝑌 ∈ 𝒫 𝑉 ↔ 𝑌 ⊆ 𝑉)
14	13	biimpri 229	. . . 4 ⊢ (𝑌 ⊆ 𝑉 → 𝑌 ∈ 𝒫 𝑉)
15	14	ad2antll 735	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → 𝑌 ∈ 𝒫 𝑉)
16		fvexd 6842	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∈ V)
17		fveq2 6827	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
18	17	ineq1d 4148	. . . . 5 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)))
19	18	fveq2d 6831	. . . 4 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))))
20		fveq2 6827	. . . . . 6 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
21	20	ineq2d 4149	. . . . 5 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))
22	21	fveq2d 6831	. . . 4 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
23		eqid 2739	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
24	19, 22, 23	ovmpog 7515	. . 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 1379	. 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 2774	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17170  HLchlt 39842  LHypclh 40476  DVecHcdvh 41570  ocHcoch 41839  joinHcdjh 41886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-djh 41887

This theorem is referenced by:  djhval2  41891  djhcl  41892  djhlj  41893  djhexmid  41903