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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 6645 | . . 3 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵):𝐵–1-1-onto→𝐵) |
3 | fvresi 6927 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥) | |
4 | fvresi 6927 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
5 | 3, 4 | breqan12d 5073 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦) ↔ 𝑥(le‘𝐾)𝑦)) |
6 | 5 | bicomd 224 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))) |
7 | 6 | rgen2 3200 | . . 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 2818 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | idlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
12 | 9, 10, 11 | islaut 37099 | . 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 709 | 1 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 I cid 5452 ↾ cres 5550 –1-1-onto→wf1o 6347 ‘cfv 6348 Basecbs 16471 lecple 16560 LAutclaut 37001 |
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 |
This theorem is referenced by: idldil 37130 |
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