Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ldilfset | Structured version Visualization version GIF version |
Description: The mapping from fiducial co-atom 𝑤 to its set of lattice dilations. (Contributed by NM, 11-May-2012.) |
Ref | Expression |
---|---|
ldilset.b | ⊢ 𝐵 = (Base‘𝐾) |
ldilset.l | ⊢ ≤ = (le‘𝐾) |
ldilset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ldilset.i | ⊢ 𝐼 = (LAut‘𝐾) |
Ref | Expression |
---|---|
ldilfset | ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V) | |
2 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) | |
3 | ldilset.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | syl6eqr 2662 | . . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
5 | fveq2 6103 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾)) | |
6 | ldilset.i | . . . . . 6 ⊢ 𝐼 = (LAut‘𝐾) | |
7 | 5, 6 | syl6eqr 2662 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼) |
8 | fveq2 6103 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
9 | ldilset.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 8, 9 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
11 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
12 | ldilset.l | . . . . . . . . 9 ⊢ ≤ = (le‘𝐾) | |
13 | 11, 12 | syl6eqr 2662 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
14 | 13 | breqd 4594 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤)) |
15 | 14 | imbi1d 330 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥))) |
16 | 10, 15 | raleqbidv 3129 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥))) |
17 | 7, 16 | rabeqbidv 3168 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}) |
18 | 4, 17 | mpteq12dv 4663 | . . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
19 | df-ldil 34408 | . . 3 ⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)})) | |
20 | fvex 6113 | . . . . 5 ⊢ (LHyp‘𝐾) ∈ V | |
21 | 3, 20 | eqeltri 2684 | . . . 4 ⊢ 𝐻 ∈ V |
22 | 21 | mptex 6390 | . . 3 ⊢ (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}) ∈ V |
23 | 18, 19, 22 | fvmpt 6191 | . 2 ⊢ (𝐾 ∈ V → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
24 | 1, 23 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 Basecbs 15695 lecple 15775 LHypclh 34288 LAutclaut 34289 LDilcldil 34404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ldil 34408 |
This theorem is referenced by: ldilset 34413 |
Copyright terms: Public domain | W3C validator |