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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 3512 | . 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
2 | lhpset.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | fveq2 6669 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | lhpset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | eqidd 2822 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤) | |
7 | fveq2 6669 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
8 | lhpset.c | . . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
9 | 7, 8 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) |
10 | fveq2 6669 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾)) | |
11 | lhpset.u | . . . . . . 7 ⊢ 1 = (1.‘𝐾) | |
12 | 10, 11 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 ) |
13 | 6, 9, 12 | breq123d 5079 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 )) |
14 | 5, 13 | rabeqbidv 3485 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
15 | df-lhyp 37123 | . . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)}) | |
16 | 4 | fvexi 6683 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16 | rabex 5234 | . . . 4 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ∈ V |
18 | 14, 15, 17 | fvmpt 6767 | . . 3 ⊢ (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
19 | 2, 18 | syl5eq 2868 | . 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 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 class class class wbr 5065 ‘cfv 6354 Basecbs 16482 1.cp1 17647 ⋖ ccvr 36397 LHypclh 37119 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-lhyp 37123 |
This theorem is referenced by: islhp 37131 |
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