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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 3462 | . 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
2 | lhpset.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | fveq2 6843 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | lhpset.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | eqtr4di 2791 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | eqidd 2734 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤) | |
7 | fveq2 6843 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾)) | |
8 | lhpset.c | . . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
9 | 7, 8 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶) |
10 | fveq2 6843 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾)) | |
11 | lhpset.u | . . . . . . 7 ⊢ 1 = (1.‘𝐾) | |
12 | 10, 11 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 ) |
13 | 6, 9, 12 | breq123d 5120 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 )) |
14 | 5, 13 | rabeqbidv 3423 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
15 | df-lhyp 38497 | . . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)}) | |
16 | 4 | fvexi 6857 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16 | rabex 5290 | . . . 4 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ∈ V |
18 | 14, 15, 17 | fvmpt 6949 | . . 3 ⊢ (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
19 | 2, 18 | eqtrid 2785 | . 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 1542 ∈ wcel 2107 {crab 3406 Vcvv 3444 class class class wbr 5106 ‘cfv 6497 Basecbs 17088 1.cp1 18318 ⋖ ccvr 37770 LHypclh 38493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-lhyp 38497 |
This theorem is referenced by: islhp 38505 |
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