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Theorem lautco 36260
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 2778 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 lautco.i . . . . 5 𝐼 = (LAut‘𝐾)
31, 2laut1o 36248 . . . 4 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
433adant3 1123 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
51, 2laut1o 36248 . . . 4 ((𝐾𝑉𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
653adant2 1122 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7 f1oco 6415 . . 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 579 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9 simpl1 1199 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾𝑉)
10 simpl2 1201 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹𝐼)
11 simpl3 1203 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺𝐼)
12 simprl 761 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
131, 2lautcl 36250 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
149, 11, 12, 13syl21anc 828 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑥) ∈ (Base‘𝐾))
15 simprr 763 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
161, 2lautcl 36250 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺𝑦) ∈ (Base‘𝐾))
179, 11, 15, 16syl21anc 828 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑦) ∈ (Base‘𝐾))
18 eqid 2778 . . . . . 6 (le‘𝐾) = (le‘𝐾)
191, 18, 2lautle 36247 . . . . 5 (((𝐾𝑉𝐹𝐼) ∧ ((𝐺𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑦) ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
209, 10, 14, 17, 19syl22anc 829 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
211, 18, 2lautle 36247 . . . . 5 (((𝐾𝑉𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
22213adantl2 1169 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
23 f1of 6393 . . . . . . 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 476 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26 fvco3 6537 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
2724, 25, 26syl2an 589 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
28 simpr 479 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29 fvco3 6537 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3024, 28, 29syl2an 589 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3127, 30breq12d 4901 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
3220, 22, 313bitr4d 303 . . 3 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
3332ralrimivva 3153 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
341, 18, 2islaut 36246 . . 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 1124 . 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 703 1 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090   class class class wbr 4888  ccom 5361  wf 6133  1-1-ontowf1o 6136  cfv 6137  Basecbs 16266  lecple 16356  LAutclaut 36148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-laut 36152
This theorem is referenced by:  ldilco  36279
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