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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idldil | Structured version Visualization version GIF version |
Description: The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
idldil.b | ⊢ 𝐵 = (Base‘𝐾) |
idldil.h | ⊢ 𝐻 = (LHyp‘𝐾) |
idldil.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
idldil | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idldil.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2731 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
3 | 1, 2 | idlaut 39431 | . . 3 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
5 | fvresi 7173 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥) | |
6 | 5 | a1d 25 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑥(le‘𝐾)𝑊 → (( I ↾ 𝐵)‘𝑥) = 𝑥)) |
7 | 6 | rgen 3062 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 (𝑥(le‘𝐾)𝑊 → (( I ↾ 𝐵)‘𝑥) = 𝑥) |
8 | 7 | a1i 11 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ 𝐵 (𝑥(le‘𝐾)𝑊 → (( I ↾ 𝐵)‘𝑥) = 𝑥)) |
9 | eqid 2731 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | idldil.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | idldil.d | . . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
12 | 1, 9, 10, 2, 11 | isldil 39445 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝐵) ∈ 𝐷 ↔ (( I ↾ 𝐵) ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥(le‘𝐾)𝑊 → (( I ↾ 𝐵)‘𝑥) = 𝑥)))) |
13 | 4, 8, 12 | mpbir2and 710 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 class class class wbr 5148 I cid 5573 ↾ cres 5678 ‘cfv 6543 Basecbs 17151 lecple 17211 LHypclh 39319 LAutclaut 39320 LDilcldil 39435 |
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-pow 5363 ax-pr 5427 ax-un 7729 |
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-pw 4604 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-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-laut 39324 df-ldil 39439 |
This theorem is referenced by: idltrn 39485 |
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