Theorem dochfval 42223

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 42222	. . . 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 6880	. . 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	eqtrid 2807	. 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 6878	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
10		dochval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11	9, 10	eqtr4di 2813	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
12	11	fveq2d 6882	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
13		dochval.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
14	12, 13	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
15	14	pweqd 4574	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
16		fveq2 6878	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = ((DIsoH‘𝐾)‘𝑊))
17		dochval.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = 𝐼)
19	18	fveq1d 6880	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘𝑦) = (𝐼‘𝑦))
20	19	sseq2d 3963	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (𝐼‘𝑦)))
21	20	rabbidv 3419	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})
22	21	fveq2d 6882	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))
23	22	fveq2d 6882	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))
24	18, 23	fveq12d 6885	. . . 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 5192	. . 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 2760	. . 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 6892	. . . . 5 ⊢ 𝑉 ∈ V
28	27	pwex 5345	. . . 4 ⊢ 𝒫 𝑉 ∈ V
29	28	mptex 7222	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) ∈ V
30	25, 26, 29	fvmpt 6986	. 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 2815	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412   ⊆ wss 3899  𝒫 cpw 4557   ↦ cmpt 5186  ‘cfv 6533  Basecbs 17301  occoc 17350  glbcglb 18398  LHypclh 40857  DVecHcdvh 41951  DIsoHcdih 42101  ocHcoch 42220

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-doch 42221

This theorem is used by:  dochval  42224  dochfN  42229