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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 39295 | . . . 4 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})) |
7 | 6 | fveq1d 6893 | . . 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 3439 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
12 | eqid 2731 | . . . 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 6905 | . . . . 5 ⊢ 𝐼 ∈ V |
14 | 13 | rabex 5332 | . . . 4 ⊢ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ∈ V |
15 | 11, 12, 14 | fvmpt 6998 | . . 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 2791 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
17 | 1, 16 | eqtrid 2783 | 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 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 Basecbs 17151 lecple 17211 LHypclh 39171 LAutclaut 39172 LDilcldil 39287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ldil 39291 |
This theorem is referenced by: isldil 39297 |
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