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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochval | Structured version Visualization version GIF version |
Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.) |
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) |
Ref | Expression |
---|---|
dochval | ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))) |
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 37513 | . . . 4 ⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))) |
10 | 9 | adantr 474 | . . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))) |
11 | 10 | fveq1d 6450 | . 2 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋)) |
12 | 7 | fvexi 6462 | . . . . . 6 ⊢ 𝑉 ∈ V |
13 | 12 | elpw2 5064 | . . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉) |
14 | 13 | biimpri 220 | . . . 4 ⊢ (𝑋 ⊆ 𝑉 → 𝑋 ∈ 𝒫 𝑉) |
15 | 14 | adantl 475 | . . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ∈ 𝒫 𝑉) |
16 | fvex 6461 | . . 3 ⊢ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ V | |
17 | sseq1 3845 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼‘𝑦) ↔ 𝑋 ⊆ (𝐼‘𝑦))) | |
18 | 17 | rabbidv 3386 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}) |
19 | 18 | fveq2d 6452 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})) |
20 | 19 | fveq2d 6452 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) |
21 | 20 | fveq2d 6452 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))) |
22 | eqid 2778 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) | |
23 | 21, 22 | fvmptg 6542 | . . 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)})))) |
24 | 15, 16, 23 | sylancl 580 | . 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)})))) |
25 | 11, 24 | eqtrd 2814 | 1 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 ⊆ wss 3792 𝒫 cpw 4379 ↦ cmpt 4967 ‘cfv 6137 Basecbs 16266 occoc 16357 glbcglb 17340 LHypclh 36147 DVecHcdvh 37241 DIsoHcdih 37391 ocHcoch 37510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-doch 37511 |
This theorem is referenced by: dochval2 37515 dochcl 37516 dochvalr 37520 dochss 37528 |
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