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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 2818 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
3 | 1, 2 | idlaut 37112 | . . 3 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
4 | 3 | adantr 481 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) |
5 | fvresi 6927 | . . . . 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 3145 | . . 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 2818 | . . 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 37126 | . 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 709 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 I cid 5452 ↾ cres 5550 ‘cfv 6348 Basecbs 16471 lecple 16560 LHypclh 37000 LAutclaut 37001 LDilcldil 37116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-laut 37005 df-ldil 37120 |
This theorem is referenced by: idltrn 37166 |
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