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Theorem lautco 39565
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 2728 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 lautco.i . . . . 5 𝐼 = (LAut‘𝐾)
31, 2laut1o 39553 . . . 4 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
433adant3 1130 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
51, 2laut1o 39553 . . . 4 ((𝐾𝑉𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
653adant2 1129 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7 f1oco 6857 . . 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 583 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9 simpl1 1189 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾𝑉)
10 simpl2 1190 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹𝐼)
11 simpl3 1191 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺𝐼)
12 simprl 770 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
131, 2lautcl 39555 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
149, 11, 12, 13syl21anc 837 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑥) ∈ (Base‘𝐾))
15 simprr 772 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
161, 2lautcl 39555 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺𝑦) ∈ (Base‘𝐾))
179, 11, 15, 16syl21anc 837 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑦) ∈ (Base‘𝐾))
18 eqid 2728 . . . . . 6 (le‘𝐾) = (le‘𝐾)
191, 18, 2lautle 39552 . . . . 5 (((𝐾𝑉𝐹𝐼) ∧ ((𝐺𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑦) ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
209, 10, 14, 17, 19syl22anc 838 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
211, 18, 2lautle 39552 . . . . 5 (((𝐾𝑉𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
22213adantl2 1165 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
23 f1of 6834 . . . . . . 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 482 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26 fvco3 6992 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
2724, 25, 26syl2an 595 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
28 simpr 484 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29 fvco3 6992 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3024, 28, 29syl2an 595 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3127, 30breq12d 5156 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
3220, 22, 313bitr4d 311 . . 3 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
3332ralrimivva 3196 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
341, 18, 2islaut 39551 . . 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 1131 . 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 712 1 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3057   class class class wbr 5143  ccom 5677  wf 6539  1-1-ontowf1o 6542  cfv 6543  Basecbs 17174  lecple 17234  LAutclaut 39453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-map 8841  df-laut 39457
This theorem is referenced by:  ldilco  39584
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