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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 2821 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
3 | 1, 2 | idlaut 37231 | . . 3 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
4 | 3 | adantr 483 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
5 | fvresi 6934 | . . . . 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 3148 | . . 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 2821 | . . 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 37245 | . 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 711 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5065 I cid 5458 ↾ cres 5556 ‘cfv 6354 Basecbs 16482 lecple 16571 LHypclh 37119 LAutclaut 37120 LDilcldil 37235 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-laut 37124 df-ldil 37239 |
This theorem is referenced by: idltrn 37285 |
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