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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isldil | Structured version Visualization version GIF version |
Description: The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (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 |
---|---|
isldil | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldilset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | ldilset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | ldilset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | ldilset.i | . . . 4 ⊢ 𝐼 = (LAut‘𝐾) | |
5 | ldilset.d | . . . 4 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ldilset 40015 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}) |
7 | 6 | eleq2d 2824 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ 𝐹 ∈ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})) |
8 | fveq1 6918 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
9 | 8 | eqeq1d 2736 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥)) |
10 | 9 | imbi2d 340 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))) |
11 | 10 | ralbidv 3180 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))) |
12 | 11 | elrab 3703 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))) |
13 | 7, 12 | bitrdi 287 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∀wral 3063 {crab 3438 class class class wbr 5169 ‘cfv 6572 Basecbs 17253 lecple 17313 LHypclh 39890 LAutclaut 39891 LDilcldil 40006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ldil 40010 |
This theorem is referenced by: ldillaut 40017 ldilval 40019 idldil 40020 ldilcnv 40021 ldilco 40022 cdleme50ldil 40454 |
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