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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 3501 | . 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
| 2 | lhpset.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
| 4 | lhpset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) | 
| 6 | eqidd 2738 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤) | |
| 7 | fveq2 6906 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
| 8 | lhpset.c | . . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) | 
| 10 | fveq2 6906 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾)) | |
| 11 | lhpset.u | . . . . . . 7 ⊢ 1 = (1.‘𝐾) | |
| 12 | 10, 11 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 ) | 
| 13 | 6, 9, 12 | breq123d 5157 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 )) | 
| 14 | 5, 13 | rabeqbidv 3455 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) | 
| 15 | df-lhyp 39990 | . . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)}) | |
| 16 | 4 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V | 
| 17 | 16 | rabex 5339 | . . . 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 2789 | . 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 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 1.cp1 18469 ⋖ ccvr 39263 LHypclh 39986 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-lhyp 39990 | 
| This theorem is referenced by: islhp 39998 | 
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