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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlaut | Structured version Visualization version GIF version |
Description: The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
idlaut.b | ⊢ 𝐵 = (Base‘𝐾) |
idlaut.i | ⊢ 𝐼 = (LAut‘𝐾) |
Ref | Expression |
---|---|
idlaut | ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6887 | . . 3 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵):𝐵–1-1-onto→𝐵) |
3 | fvresi 7193 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥) | |
4 | fvresi 7193 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
5 | 3, 4 | breqan12d 5164 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦) ↔ 𝑥(le‘𝐾)𝑦)) |
6 | 5 | bicomd 223 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))) |
7 | 6 | rgen2 3197 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦)) |
8 | 7 | a1i 11 | . 2 ⊢ (𝐾 ∈ 𝐴 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))) |
9 | idlaut.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
10 | eqid 2735 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | idlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
12 | 9, 10, 11 | islaut 40066 | . 2 ⊢ (𝐾 ∈ 𝐴 → (( I ↾ 𝐵) ∈ 𝐼 ↔ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))))) |
13 | 2, 8, 12 | mpbir2and 713 | 1 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 I cid 5582 ↾ cres 5691 –1-1-onto→wf1o 6562 ‘cfv 6563 Basecbs 17245 lecple 17305 LAutclaut 39968 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-laut 39972 |
This theorem is referenced by: idldil 40097 |
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