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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnset | Structured version Visualization version GIF version |
Description: The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.) |
Ref | Expression |
---|---|
llnset.b | ⊢ 𝐵 = (Base‘𝐾) |
llnset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
llnset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
llnset.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
llnset | ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V) | |
2 | llnset.n | . . 3 ⊢ 𝑁 = (LLines‘𝐾) | |
3 | fveq2 6645 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | llnset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | eqtr4di 2851 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | fveq2 6645 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
7 | llnset.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 6, 7 | eqtr4di 2851 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
9 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
10 | llnset.c | . . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
11 | 9, 10 | eqtr4di 2851 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) |
12 | 11 | breqd 5041 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥 ↔ 𝑝𝐶𝑥)) |
13 | 8, 12 | rexeqbidv 3355 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥)) |
14 | 5, 13 | rabeqbidv 3433 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}) |
15 | df-llines 36794 | . . . 4 ⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}) | |
16 | 4 | fvexi 6659 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16 | rabex 5199 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ∈ V |
18 | 14, 15, 17 | fvmpt 6745 | . . 3 ⊢ (𝐾 ∈ V → (LLines‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}) |
19 | 2, 18 | syl5eq 2845 | . 2 ⊢ (𝐾 ∈ V → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 ⋖ ccvr 36558 Atomscatm 36559 LLinesclln 36787 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-llines 36794 |
This theorem is referenced by: islln 36802 |
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