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Theorem lautco 40100
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 2736 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 lautco.i . . . . 5 𝐼 = (LAut‘𝐾)
31, 2laut1o 40088 . . . 4 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
433adant3 1132 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
51, 2laut1o 40088 . . . 4 ((𝐾𝑉𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
653adant2 1131 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7 f1oco 6870 . . 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 584 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9 simpl1 1191 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾𝑉)
10 simpl2 1192 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹𝐼)
11 simpl3 1193 . . . . . 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 40090 . . . . . 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 40090 . . . . . 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 2736 . . . . . 6 (le‘𝐾) = (le‘𝐾)
191, 18, 2lautle 40087 . . . . 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 40087 . . . . 5 (((𝐾𝑉𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
22213adantl2 1167 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
23 f1of 6847 . . . . . . 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 7007 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
2724, 25, 26syl2an 596 . . . . 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 7007 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3024, 28, 29syl2an 596 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3127, 30breq12d 5155 . . . 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 3201 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
341, 18, 2islaut 40086 . . 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 1133 . 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 713 1 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142  ccom 5688  wf 6556  1-1-ontowf1o 6559  cfv 6560  Basecbs 17248  lecple 17305  LAutclaut 39988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-laut 39992
This theorem is referenced by:  ldilco  40119
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