Theorem djavalN 38265

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 38264	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
7	6	adantr 483	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
8	7	oveqd 7167	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))𝑌))
9	2	fvexi 6678	. . . . . 6 ⊢ 𝑇 ∈ V
10	9	elpw2 5240	. . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑇 ↔ 𝑋 ⊆ 𝑇)
11	10	biimpri 230	. . . 4 ⊢ (𝑋 ⊆ 𝑇 → 𝑋 ∈ 𝒫 𝑇)
12	11	ad2antrl 726	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → 𝑋 ∈ 𝒫 𝑇)
13	9	elpw2 5240	. . . . 5 ⊢ (𝑌 ∈ 𝒫 𝑇 ↔ 𝑌 ⊆ 𝑇)
14	13	biimpri 230	. . . 4 ⊢ (𝑌 ⊆ 𝑇 → 𝑌 ∈ 𝒫 𝑇)
15	14	ad2antll 727	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → 𝑌 ∈ 𝒫 𝑇)
16		fvexd 6679	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∈ V)
17		fveq2 6664	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
18	17	ineq1d 4187	. . . . 5 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)))
19	18	fveq2d 6668	. . . 4 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))))
20		fveq2 6664	. . . . . 6 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
21	20	ineq2d 4188	. . . . 5 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))
22	21	fveq2d 6668	. . . 4 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
23		eqid 2821	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
24	19, 22, 23	ovmpog 7303	. . 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 1367	. 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 2856	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  HLchlt 36480  LHypclh 37114  LTrncltrn 37231  DIsoAcdia 38158  ocAcocaN 38249  vAcdjaN 38261

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-djaN 38262

This theorem is referenced by:  djaclN  38266  djajN  38267