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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 3212 | . 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
2 | lvolset.v | . . 3 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | lvolset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | syl6eqr 2674 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | fveq2 6191 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾)) | |
7 | lvolset.p | . . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾) | |
8 | 6, 7 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃) |
9 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
10 | lvolset.c | . . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) |
12 | 11 | breqd 4664 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥)) |
13 | 8, 12 | rexeqbidv 3153 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥)) |
14 | 5, 13 | rabeqbidv 3195 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
15 | df-lvols 34786 | . . . 4 ⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥}) | |
16 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝐾) ∈ V | |
17 | 4, 16 | eqeltri 2697 | . . . . 5 ⊢ 𝐵 ∈ V |
18 | 17 | rabex 4813 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥} ∈ V |
19 | 14, 15, 18 | fvmpt 6282 | . . 3 ⊢ (𝐾 ∈ V → (LVols‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
20 | 2, 19 | syl5eq 2668 | . 2 ⊢ (𝐾 ∈ V → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
21 | 1, 20 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 ⋖ ccvr 34549 LPlanesclpl 34778 LVolsclvol 34779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-lvols 34786 |
This theorem is referenced by: islvol 34859 |
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