Theorem dochval 42166

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 42165	. . . 4 ⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
10	9	adantr 486	. . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
11	10	fveq1d 6890	. 2 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋))
12	7	fvexi 6902	. . . . 5 ⊢ 𝑉 ∈ V
13	12	elpw2 5310	. . . 4 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉)
14	13	bilanri 512	. . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ∈ 𝒫 𝑉)
15		fvex 6901	. . 3 ⊢ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ V
16		sseq1 3965	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼‘𝑦) ↔ 𝑋 ⊆ (𝐼‘𝑦)))
17	16	rabbidv 3426	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})
18	17	fveq2d 6892	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))
19	18	fveq2d 6892	. . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))
20	19	fveq2d 6892	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))
21		eqid 2766	. . . 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 6994	. . 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 598	. 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 2801	1 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458   ⊆ wss 3908  𝒫 cpw 4567   ↦ cmpt 5197  ‘cfv 6543  Basecbs 17294  occoc 17343  glbcglb 18391  LHypclh 40799  DVecHcdvh 41893  DIsoHcdih 42043  ocHcoch 42162

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-doch 42163

This theorem is used by:  dochval2  42167  dochcl  42168  dochvalr  42172  dochss  42180