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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 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | laut1o.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
4 | 1, 2, 3 | islaut 37213 | . 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 501 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼) → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5058 –1-1-onto→wf1o 6348 ‘cfv 6349 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-laut 37119 |
This theorem is referenced by: laut11 37216 lautcl 37217 lautcnvclN 37218 lautcnvle 37219 lautcnv 37220 lautcvr 37222 lautj 37223 lautm 37224 lautco 37227 ldil1o 37242 ltrn1o 37254 |
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