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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 2737 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
| 3 | 1, 2 | idlaut 40098 | . . 3 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) | 
| 4 | 3 | adantr 480 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ (LAut‘𝐾)) | 
| 5 | fvresi 7193 | . . . . 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 3063 | . . 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 2737 | . . 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 40112 | . 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 713 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 I cid 5577 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 lecple 17304 LHypclh 39986 LAutclaut 39987 LDilcldil 40102 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-laut 39991 df-ldil 40106 | 
| This theorem is referenced by: idltrn 40152 | 
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