Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > laut1o | Structured version Visualization version GIF version |
Description: A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.) |
Ref | Expression |
---|---|
laut1o.b | ⊢ 𝐵 = (Base‘𝐾) |
laut1o.i | ⊢ 𝐼 = (LAut‘𝐾) |
Ref | Expression |
---|---|
laut1o | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼) → 𝐹:𝐵–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | laut1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | laut1o.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
4 | 1, 2, 3 | islaut 34387 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝐹 ∈ 𝐼 ↔ (𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(le‘𝐾)𝑦 ↔ (𝐹‘𝑥)(le‘𝐾)(𝐹‘𝑦))))) |
5 | 4 | simprbda 651 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼) → 𝐹:𝐵–1-1-onto→𝐵) |
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 –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: laut11 34390 lautcl 34391 lautcnvclN 34392 lautcnvle 34393 lautcnv 34394 lautcvr 34396 lautj 34397 lautm 34398 lautco 34401 ldil1o 34416 ltrn1o 34428 |
Copyright terms: Public domain | W3C validator |