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Theorem lautco 34401
 Description: The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
Hypothesis
Ref Expression
lautco.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautco ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)

Proof of Theorem lautco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 lautco.i . . . . 5 𝐼 = (LAut‘𝐾)
31, 2laut1o 34389 . . . 4 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
433adant3 1074 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
51, 2laut1o 34389 . . . 4 ((𝐾𝑉𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
653adant2 1073 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7 f1oco 6072 . . 3 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾)) → (𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
84, 6, 7syl2anc 691 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9 simpl1 1057 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾𝑉)
10 simpl2 1058 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹𝐼)
11 simpl3 1059 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺𝐼)
12 simprl 790 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
131, 2lautcl 34391 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
149, 11, 12, 13syl21anc 1317 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑥) ∈ (Base‘𝐾))
15 simprr 792 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
161, 2lautcl 34391 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺𝑦) ∈ (Base‘𝐾))
179, 11, 15, 16syl21anc 1317 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑦) ∈ (Base‘𝐾))
18 eqid 2610 . . . . . 6 (le‘𝐾) = (le‘𝐾)
191, 18, 2lautle 34388 . . . . 5 (((𝐾𝑉𝐹𝐼) ∧ ((𝐺𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑦) ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
209, 10, 14, 17, 19syl22anc 1319 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
211, 18, 2lautle 34388 . . . . 5 (((𝐾𝑉𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
22213adantl2 1211 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
23 f1of 6050 . . . . . . 7 (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
246, 23syl 17 . . . . . 6 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
25 simpl 472 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26 fvco3 6185 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
2724, 25, 26syl2an 493 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
28 simpr 476 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29 fvco3 6185 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3024, 28, 29syl2an 493 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3127, 30breq12d 4596 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
3220, 22, 313bitr4d 299 . . 3 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
3332ralrimivva 2954 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
341, 18, 2islaut 34387 . . 3 (𝐾𝑉 → ((𝐹𝐺) ∈ 𝐼 ↔ ((𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))))
35343ad2ant1 1075 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → ((𝐹𝐺) ∈ 𝐼 ↔ ((𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))))
368, 33, 35mpbir2and 959 1 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   ∘ ccom 5042  ⟶wf 5800  –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:  ldilco  34420
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