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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 41333 | . . . 4 ⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))) |
10 | 9 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))) |
11 | 10 | fveq1d 6909 | . 2 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋)) |
12 | 7 | fvexi 6921 | . . . . . 6 ⊢ 𝑉 ∈ V |
13 | 12 | elpw2 5340 | . . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉) |
14 | 13 | biimpri 228 | . . . 4 ⊢ (𝑋 ⊆ 𝑉 → 𝑋 ∈ 𝒫 𝑉) |
15 | 14 | adantl 481 | . . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ∈ 𝒫 𝑉) |
16 | fvex 6920 | . . 3 ⊢ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ V | |
17 | sseq1 4021 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼‘𝑦) ↔ 𝑋 ⊆ (𝐼‘𝑦))) | |
18 | 17 | rabbidv 3441 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}) |
19 | 18 | fveq2d 6911 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})) |
20 | 19 | fveq2d 6911 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) |
21 | 20 | fveq2d 6911 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))) |
22 | eqid 2735 | . . . 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 7014 | . . 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 586 | . 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 2775 | 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 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 ‘cfv 6563 Basecbs 17245 occoc 17306 glbcglb 18368 LHypclh 39967 DVecHcdvh 41061 DIsoHcdih 41211 ocHcoch 41330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-doch 41331 |
This theorem is referenced by: dochval2 41335 dochcl 41336 dochvalr 41340 dochss 41348 |
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