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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 6174 | . . 3 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵):𝐵–1-1-onto→𝐵) |
3 | fvresi 6439 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥) | |
4 | fvresi 6439 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
5 | 3, 4 | breqan12d 4669 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦) ↔ 𝑥(le‘𝐾)𝑦)) |
6 | 5 | bicomd 213 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(le‘𝐾)𝑦 ↔ (( I ↾ 𝐵)‘𝑥)(le‘𝐾)(( I ↾ 𝐵)‘𝑦))) |
7 | 6 | rgen2a 2977 | . . 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 2622 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | idlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
12 | 9, 10, 11 | islaut 35369 | . 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 957 | 1 ⊢ (𝐾 ∈ 𝐴 → ( I ↾ 𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 I cid 5023 ↾ cres 5116 –1-1-onto→wf1o 5887 ‘cfv 5888 Basecbs 15857 lecple 15948 LAutclaut 35271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-laut 35275 |
This theorem is referenced by: idldil 35400 |
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