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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 6086 | . . 3 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵):𝐵–1-1-onto→𝐵) |
3 | fvresi 6344 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥) | |
4 | fvresi 6344 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
5 | 3, 4 | breqan12d 4599 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦) ↔ 𝑥(le‘𝐾)𝑦)) |
6 | 5 | bicomd 212 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))) |
7 | 6 | rgen2a 2960 | . . 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 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | idlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
12 | 9, 10, 11 | islaut 34387 | . 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 959 | 1 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 I cid 4948 ↾ cres 5040 –1-1-onto→wf1o 5803 ‘cfv 5804 Basecbs 15695 lecple 15775 LAutclaut 34289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-laut 34293 |
This theorem is referenced by: idldil 34418 |
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