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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 2778 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
3 | 1, 2 | idlaut 36259 | . . 3 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
4 | 3 | adantr 474 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
5 | fvresi 6708 | . . . . 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 3104 | . . 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 2778 | . . 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 36273 | . 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 703 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 class class class wbr 4888 I cid 5262 ↾ cres 5359 ‘cfv 6137 Basecbs 16266 lecple 16356 LHypclh 36147 LAutclaut 36148 LDilcldil 36263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-laut 36152 df-ldil 36267 |
This theorem is referenced by: idltrn 36313 |
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