Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 6647 | . . 3 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵):𝐵–1-1-onto→𝐵) |
3 | fvresi 6930 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥) | |
4 | fvresi 6930 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
5 | 3, 4 | breqan12d 5075 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦) ↔ 𝑥(le‘𝐾)𝑦)) |
6 | 5 | bicomd 225 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))) |
7 | 6 | rgen2 3203 | . . 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 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | idlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
12 | 9, 10, 11 | islaut 37213 | . 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 711 | 1 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5059 I cid 5454 ↾ cres 5552 –1-1-onto→wf1o 6349 ‘cfv 6350 Basecbs 16477 lecple 16566 LAutclaut 37115 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-laut 37119 |
This theorem is referenced by: idldil 37244 |
Copyright terms: Public domain | W3C validator |