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Theorem lautcnv 34856
Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
Hypothesis
Ref Expression
lautcnv.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautcnv ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)

Proof of Theorem lautcnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lautcnv.i . . . 4 𝐼 = (LAut‘𝐾)
31, 2laut1o 34851 . . 3 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4 f1ocnv 6106 . . 3 (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
53, 4syl 17 . 2 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6 eqid 2621 . . . 4 (le‘𝐾) = (le‘𝐾)
71, 6, 2lautcnvle 34855 . . 3 (((𝐾𝑉𝐹𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
87ralrimivva 2965 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
91, 6, 2islaut 34849 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
109adantr 481 . 2 ((𝐾𝑉𝐹𝐼) → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
115, 8, 10mpbir2and 956 1 ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4613  ccnv 5073  1-1-ontowf1o 5846  cfv 5847  Basecbs 15781  lecple 15869  LAutclaut 34751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-laut 34755
This theorem is referenced by:  ldilcnv  34881
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