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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 3455 | . 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V) | |
2 | fveq2 6538 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) | |
3 | ldilset.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | syl6eqr 2849 | . . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
5 | fveq2 6538 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾)) | |
6 | ldilset.i | . . . . . 6 ⊢ 𝐼 = (LAut‘𝐾) | |
7 | 5, 6 | syl6eqr 2849 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼) |
8 | fveq2 6538 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
9 | ldilset.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 8, 9 | syl6eqr 2849 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
11 | fveq2 6538 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
12 | ldilset.l | . . . . . . . . 9 ⊢ ≤ = (le‘𝐾) | |
13 | 11, 12 | syl6eqr 2849 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
14 | 13 | breqd 4973 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤)) |
15 | 14 | imbi1d 343 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥))) |
16 | 10, 15 | raleqbidv 3361 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥))) |
17 | 7, 16 | rabeqbidv 3430 | . . . 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 5045 | . . 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 36771 | . . 3 ⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)})) | |
20 | 18, 19, 3 | mptfvmpt 6856 | . 2 ⊢ (𝐾 ∈ V → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
21 | 1, 20 | 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 1522 ∈ wcel 2081 ∀wral 3105 {crab 3109 Vcvv 3437 class class class wbr 4962 ↦ cmpt 5041 ‘cfv 6225 Basecbs 16312 lecple 16401 LHypclh 36651 LAutclaut 36652 LDilcldil 36767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ldil 36771 |
This theorem is referenced by: ldilset 36776 |
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