Theorem dochfval 39291

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
dochfval	⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))

Distinct variable groups:   𝑦,𝐵   𝑥,𝑦,𝐾   𝑥,𝑉   𝑥,𝑊,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝑈(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝑁(𝑥,𝑦)   ⊥ (𝑥,𝑦)   𝑉(𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem dochfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochval.n	. . 3 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊)
2		dochval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		dochval.g	. . . . 5 ⊢ 𝐺 = (glb‘𝐾)
4		dochval.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
5		dochval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6	2, 3, 4, 5	dochffval 39290	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (ocH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
7	6	fveq1d 6758	. . 3 ⊢ (𝐾 ∈ 𝑋 → ((ocH‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
8	1, 7	syl5eq 2791	. 2 ⊢ (𝐾 ∈ 𝑋 → 𝑁 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊))
9		fveq2 6756	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
10		dochval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11	9, 10	eqtr4di 2797	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
12	11	fveq2d 6760	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
13		dochval.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
14	12, 13	eqtr4di 2797	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
15	14	pweqd 4549	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
16		fveq2 6756	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = ((DIsoH‘𝐾)‘𝑊))
17		dochval.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2797	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = 𝐼)
19	18	fveq1d 6758	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘𝑦) = (𝐼‘𝑦))
20	19	sseq2d 3949	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (𝐼‘𝑦)))
21	20	rabbidv 3404	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})
22	21	fveq2d 6760	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))
23	22	fveq2d 6760	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))
24	18, 23	fveq12d 6763	. . . 4 ⊢ (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))
25	15, 24	mpteq12dv 5161	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
26		eqid 2738	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))
27	13	fvexi 6770	. . . . 5 ⊢ 𝑉 ∈ V
28	27	pwex 5298	. . . 4 ⊢ 𝒫 𝑉 ∈ V
29	28	mptex 7081	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) ∈ V
30	25, 26, 29	fvmpt 6857	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))‘𝑊) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
31	8, 30	sylan9eq 2799	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883  𝒫 cpw 4530   ↦ cmpt 5153  ‘cfv 6418  Basecbs 16840  occoc 16896  glbcglb 17943  LHypclh 37925  DVecHcdvh 39019  DIsoHcdih 39169  ocHcoch 39288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-doch 39289

This theorem is referenced by:  dochval  39292  dochfN  39297