Theorem dochfval 41332

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 41331	. . . 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 6828	. . 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 2776	. 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 6826	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
10		dochval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11	9, 10	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
12	11	fveq2d 6830	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
13		dochval.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
14	12, 13	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
15	14	pweqd 4570	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
16		fveq2 6826	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = ((DIsoH‘𝐾)‘𝑊))
17		dochval.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoH‘𝐾)‘𝑤) = 𝐼)
19	18	fveq1d 6828	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((DIsoH‘𝐾)‘𝑤)‘𝑦) = (𝐼‘𝑦))
20	19	sseq2d 3970	. . . . . . . 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 6830	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))
23	22	fveq2d 6830	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))
24	18, 23	fveq12d 6833	. . . 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 5182	. . 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 2729	. . 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 6840	. . . . 5 ⊢ 𝑉 ∈ V
28	27	pwex 5322	. . . 4 ⊢ 𝒫 𝑉 ∈ V
29	28	mptex 7163	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) ∈ V
30	25, 26, 29	fvmpt 6934	. 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 2784	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3396   ⊆ wss 3905  𝒫 cpw 4553   ↦ cmpt 5176  ‘cfv 6486  Basecbs 17138  occoc 17187  glbcglb 18234  LHypclh 39966  DVecHcdvh 41060  DIsoHcdih 41210  ocHcoch 41329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-doch 41330

This theorem is referenced by:  dochval  41333  dochfN  41338