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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 3512 | . 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V) | |
2 | fveq2 6664 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) | |
3 | ldilset.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | syl6eqr 2874 | . . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
5 | fveq2 6664 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾)) | |
6 | ldilset.i | . . . . . 6 ⊢ 𝐼 = (LAut‘𝐾) | |
7 | 5, 6 | syl6eqr 2874 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼) |
8 | fveq2 6664 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
9 | ldilset.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 8, 9 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
11 | fveq2 6664 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
12 | ldilset.l | . . . . . . . . 9 ⊢ ≤ = (le‘𝐾) | |
13 | 11, 12 | syl6eqr 2874 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
14 | 13 | breqd 5069 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤)) |
15 | 14 | imbi1d 344 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥))) |
16 | 10, 15 | raleqbidv 3401 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥))) |
17 | 7, 16 | rabeqbidv 3485 | . . . 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 5143 | . . 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 37234 | . . 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 6984 | . 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 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 Vcvv 3494 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 Basecbs 16477 lecple 16566 LHypclh 37114 LAutclaut 37115 LDilcldil 37230 |
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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ldil 37234 |
This theorem is referenced by: ldilset 37239 |
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