Theorem dochffval 41719

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)

Assertion

Ref	Expression
dochffval	⊢ (𝐾 ∈ 𝑉 → (ocH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))

Distinct variable groups:   𝑦,𝐵   𝑤,𝐻   𝑥,𝑤,𝑦,𝐾

Allowed substitution hints:   𝐵(𝑥,𝑤)   𝐺(𝑥,𝑦,𝑤)   𝐻(𝑥,𝑦)   ⊥ (𝑥,𝑦,𝑤)   𝑉(𝑥,𝑦,𝑤)

Proof of Theorem dochffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6842	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dochval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2790	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
6	5	fveq1d 6844	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
7	6	fveq2d 6846	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
8	7	pweqd 4573	. . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) = 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)))
9		fveq2 6842	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DIsoH‘𝑘) = (DIsoH‘𝐾))
10	9	fveq1d 6844	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DIsoH‘𝑘)‘𝑤) = ((DIsoH‘𝐾)‘𝑤))
11		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
12		dochval.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
13	11, 12	eqtr4di 2790	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
14		fveq2 6842	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
15		dochval.g	. . . . . . . . 9 ⊢ 𝐺 = (glb‘𝐾)
16	14, 15	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
17		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
18		dochval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
19	17, 18	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
20	10	fveq1d 6844	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘𝑦) = (((DIsoH‘𝐾)‘𝑤)‘𝑦))
21	20	sseq2d 3968	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦) ↔ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)))
22	19, 21	rabeqbidv 3419	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})
23	16, 22	fveq12d 6849	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))
24	13, 23	fveq12d 6849	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))
25	10, 24	fveq12d 6849	. . . . 5 ⊢ (𝑘 = 𝐾 → (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))) = (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))
26	8, 25	mpteq12dv 5187	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))) = (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))
27	4, 26	mpteq12dv 5187	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
28		df-doch 41718	. . 3 ⊢ ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))
29	27, 28, 3	mptfvmpt 7184	. 2 ⊢ (𝐾 ∈ V → (ocH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
30	1, 29	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (ocH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556   ↦ cmpt 5181  ‘cfv 6500  Basecbs 17148  occoc 17197  glbcglb 18245  LHypclh 40354  DVecHcdvh 41448  DIsoHcdih 41598  ocHcoch 41717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-doch 41718

This theorem is referenced by:  dochfval  41720