Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lautco Structured version   Visualization version   GIF version

Theorem lautco 34898
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 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 lautco.i . . . . 5 𝐼 = (LAut‘𝐾)
31, 2laut1o 34886 . . . 4 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
433adant3 1079 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
51, 2laut1o 34886 . . . 4 ((𝐾𝑉𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
653adant2 1078 . . 3 ((𝐾𝑉𝐹𝐼𝐺𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7 f1oco 6121 . . 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 692 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9 simpl1 1062 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾𝑉)
10 simpl2 1063 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹𝐼)
11 simpl3 1064 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺𝐼)
12 simprl 793 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
131, 2lautcl 34888 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
149, 11, 12, 13syl21anc 1322 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑥) ∈ (Base‘𝐾))
15 simprr 795 . . . . . 6 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
161, 2lautcl 34888 . . . . . 6 (((𝐾𝑉𝐺𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺𝑦) ∈ (Base‘𝐾))
179, 11, 15, 16syl21anc 1322 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺𝑦) ∈ (Base‘𝐾))
18 eqid 2621 . . . . . 6 (le‘𝐾) = (le‘𝐾)
191, 18, 2lautle 34885 . . . . 5 (((𝐾𝑉𝐹𝐼) ∧ ((𝐺𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑦) ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
209, 10, 14, 17, 19syl22anc 1324 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐺𝑥)(le‘𝐾)(𝐺𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
211, 18, 2lautle 34885 . . . . 5 (((𝐾𝑉𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
22213adantl2 1216 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺𝑥)(le‘𝐾)(𝐺𝑦)))
23 f1of 6099 . . . . . . 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 473 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26 fvco3 6237 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
2724, 25, 26syl2an 494 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
28 simpr 477 . . . . . 6 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29 fvco3 6237 . . . . . 6 ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3024, 28, 29syl2an 494 . . . . 5 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3127, 30breq12d 4631 . . . 4 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦) ↔ (𝐹‘(𝐺𝑥))(le‘𝐾)(𝐹‘(𝐺𝑦))))
3220, 22, 313bitr4d 300 . . 3 (((𝐾𝑉𝐹𝐼𝐺𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
3332ralrimivva 2966 . 2 ((𝐾𝑉𝐹𝐼𝐺𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹𝐺)‘𝑥)(le‘𝐾)((𝐹𝐺)‘𝑦)))
341, 18, 2islaut 34884 . . 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 1080 . 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 956 1 ((𝐾𝑉𝐹𝐼𝐺𝐼) → (𝐹𝐺) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618  ccom 5083  wf 5848  1-1-ontowf1o 5851  cfv 5852  Basecbs 15792  lecple 15880  LAutclaut 34786
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-laut 34790
This theorem is referenced by:  ldilco  34917
  Copyright terms: Public domain W3C validator