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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpset | Structured version Visualization version GIF version |
Description: The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.) |
Ref | Expression |
---|---|
lhpset.b | ⊢ 𝐵 = (Base‘𝐾) |
lhpset.u | ⊢ 1 = (1.‘𝐾) |
lhpset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
lhpset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpset | ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3499 | . 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
2 | lhpset.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | lhpset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | eqidd 2736 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤) | |
7 | fveq2 6907 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
8 | lhpset.c | . . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
9 | 7, 8 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) |
10 | fveq2 6907 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾)) | |
11 | lhpset.u | . . . . . . 7 ⊢ 1 = (1.‘𝐾) | |
12 | 10, 11 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 ) |
13 | 6, 9, 12 | breq123d 5162 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 )) |
14 | 5, 13 | rabeqbidv 3452 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
15 | df-lhyp 39971 | . . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)}) | |
16 | 4 | fvexi 6921 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16 | rabex 5345 | . . . 4 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ∈ V |
18 | 14, 15, 17 | fvmpt 7016 | . . 3 ⊢ (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
19 | 2, 18 | eqtrid 2787 | . 2 ⊢ (𝐾 ∈ V → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 1.cp1 18482 ⋖ ccvr 39244 LHypclh 39967 |
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-sep 5302 ax-nul 5312 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-rab 3434 df-v 3480 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-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-iota 6516 df-fun 6565 df-fv 6571 df-lhyp 39971 |
This theorem is referenced by: islhp 39979 |
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