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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 35657 | . . . 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 6105 | . 2 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = ((𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))‘𝑋)) |
12 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝑈) ∈ V | |
13 | 7, 12 | eqeltri 2684 | . . . . . 6 ⊢ 𝑉 ∈ V |
14 | 13 | elpw2 4755 | . . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉) |
15 | 14 | biimpri 217 | . . . 4 ⊢ (𝑋 ⊆ 𝑉 → 𝑋 ∈ 𝒫 𝑉) |
16 | 15 | adantl 481 | . . 3 ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ∈ 𝒫 𝑉) |
17 | fvex 6113 | . . 3 ⊢ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ V | |
18 | sseq1 3589 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ (𝐼‘𝑦) ↔ 𝑋 ⊆ (𝐼‘𝑦))) | |
19 | 18 | rabbidv 3164 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)} = {𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}) |
20 | 19 | fveq2d 6107 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}) = (𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})) |
21 | 20 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})) = ( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) |
22 | 21 | fveq2d 6107 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))) |
23 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))) | |
24 | 22, 23 | fvmptg 6189 | . . 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)})))) |
25 | 16, 17, 24 | sylancl 693 | . 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)})))) |
26 | 11, 25 | eqtrd 2644 | 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 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ↦ cmpt 4643 ‘cfv 5804 Basecbs 15695 occoc 15776 glbcglb 16766 LHypclh 34288 DVecHcdvh 35385 DIsoHcdih 35535 ocHcoch 35654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-doch 35655 |
This theorem is referenced by: dochval2 35659 dochcl 35660 dochvalr 35664 dochss 35672 |
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