Theorem dochval 41850

Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)

Hypotheses

Ref	Expression
dochval.b	⊢ 𝐵 = (Base‘𝐾)
dochval.g	⊢ 𝐺 = (glb‘𝐾)
dochval.o	⊢ ⊥ = (oc‘𝐾)
dochval.h	⊢ 𝐻 = (LHyp‘𝐾)
dochval.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dochval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochval.v	⊢ 𝑉 = (Base‘𝑈)
dochval.n	⊢ 𝑁 = ((ocH‘𝐾)‘𝑊)

Assertion

Ref	Expression
dochval	⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))

Distinct variable groups:   𝑦,𝐵   𝑦,𝐾   𝑦,𝑊   𝑦,𝑋

Allowed substitution hints:   𝑈(𝑦)   𝐺(𝑦)   𝐻(𝑦)   𝐼(𝑦)   𝑁(𝑦)   ⊥ (𝑦)   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem dochval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		dochval.g	. . . . 5 ⊢ 𝐺 = (glb‘𝐾)
3		dochval.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
4		dochval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		dochval.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6		dochval.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		dochval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		dochval.n	. . . . 5 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	dochfval 41849	. . . 4 ⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
10	9	adantr 481	. . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
11	10	fveq1d 6836	. 2 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋))
12	7	fvexi 6848	. . . . 5 ⊢ 𝑉 ∈ V
13	12	elpw2 5269	. . . 4 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉)
14	13	bilanri 507	. . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ∈ 𝒫 𝑉)
15		fvex 6847	. . 3 ⊢ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ V
16		sseq1 3947	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼‘𝑦) ↔ 𝑋 ⊆ (𝐼‘𝑦)))
17	16	rabbidv 3399	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})
18	17	fveq2d 6838	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))
19	18	fveq2d 6838	. . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))
20	19	fveq2d 6838	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))
21		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))
22	20, 21	fvmptg 6940	. . 3 ⊢ ((𝑋 ∈ 𝒫 𝑉 ∧ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ V) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))
23	14, 15, 22	sylancl 592	. 2 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))
24	11, 23	eqtrd 2775	1 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ⊆ wss 3890  𝒫 cpw 4536   ↦ cmpt 5160  ‘cfv 6492  Basecbs 17177  occoc 17226  glbcglb 18274  LHypclh 40483  DVecHcdvh 41577  DIsoHcdih 41727  ocHcoch 41846

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-doch 41847

This theorem is referenced by:  dochval2  41851  dochcl  41852  dochvalr  41856  dochss  41864