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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldilset | Structured version Visualization version GIF version |
Description: The set of lattice dilations for a fiducial co-atom 𝑊. (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) |
ldilset.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ldilset | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldilset.d | . 2 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
2 | ldilset.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | ldilset.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
4 | ldilset.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | ldilset.i | . . . . 5 ⊢ 𝐼 = (LAut‘𝐾) | |
6 | 2, 3, 4, 5 | ldilfset 40091 | . . . 4 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
7 | 6 | fveq1d 6909 | . . 3 ⊢ (𝐾 ∈ 𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})‘𝑊)) |
8 | breq2 5152 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊)) | |
9 | 8 | imbi1d 341 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥))) |
10 | 9 | ralbidv 3176 | . . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥))) |
11 | 10 | rabbidv 3441 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
12 | eqid 2735 | . . . 4 ⊢ (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}) | |
13 | 5 | fvexi 6921 | . . . . 5 ⊢ 𝐼 ∈ V |
14 | 13 | rabex 5345 | . . . 4 ⊢ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ∈ V |
15 | 11, 12, 14 | fvmpt 7016 | . . 3 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
16 | 7, 15 | sylan9eq 2795 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
17 | 1, 16 | eqtrid 2787 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 Basecbs 17245 lecple 17305 LHypclh 39967 LAutclaut 39968 LDilcldil 40083 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ldil 40087 |
This theorem is referenced by: isldil 40093 |
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