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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolset | Structured version Visualization version GIF version |
Description: The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
lvolset.b | ⊢ 𝐵 = (Base‘𝐾) |
lvolset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
lvolset.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
lvolset.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
lvolset | ⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3463 | . 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
2 | lvolset.v | . . 3 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | fveq2 6842 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | lvolset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | eqtr4di 2794 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | fveq2 6842 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾)) | |
7 | lvolset.p | . . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾) | |
8 | 6, 7 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃) |
9 | fveq2 6842 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
10 | lvolset.c | . . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
11 | 9, 10 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) |
12 | 11 | breqd 5116 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥)) |
13 | 8, 12 | rexeqbidv 3320 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥)) |
14 | 5, 13 | rabeqbidv 3424 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
15 | df-lvols 37953 | . . . 4 ⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥}) | |
16 | 4 | fvexi 6856 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16 | rabex 5289 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥} ∈ V |
18 | 14, 15, 17 | fvmpt 6948 | . . 3 ⊢ (𝐾 ∈ V → (LVols‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
19 | 2, 18 | eqtrid 2788 | . 2 ⊢ (𝐾 ∈ V → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 {crab 3407 Vcvv 3445 class class class wbr 5105 ‘cfv 6496 Basecbs 17082 ⋖ ccvr 37714 LPlanesclpl 37945 LVolsclvol 37946 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-lvols 37953 |
This theorem is referenced by: islvol 38026 |
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