Theorem lautco 37235

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	⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺) ∈ 𝐼)

Proof of Theorem lautco

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautco.i	. . . . 5 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 37223	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4	3	3adant3 1128	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	1, 2	laut1o 37223	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1127	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1oco 6639	. . 3 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾)) → (𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8	4, 6, 7	syl2anc 586	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		simpl1 1187	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾 ∈ 𝑉)
10		simpl2 1188	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹 ∈ 𝐼)
11		simpl3 1189	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺 ∈ 𝐼)
12		simprl 769	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
13	1, 2	lautcl 37225	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
14	9, 11, 12, 13	syl21anc 835	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑥) ∈ (Base‘𝐾))
15		simprr 771	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
16	1, 2	lautcl 37225	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺‘𝑦) ∈ (Base‘𝐾))
17	9, 11, 15, 16	syl21anc 835	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑦) ∈ (Base‘𝐾))
18		eqid 2823	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
19	1, 18, 2	lautle 37222	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ ((𝐺‘𝑥) ∈ (Base‘𝐾) ∧ (𝐺‘𝑦) ∈ (Base‘𝐾))) → ((𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦) ↔ (𝐹‘(𝐺‘𝑥))(le‘𝐾)(𝐹‘(𝐺‘𝑦))))
20	9, 10, 14, 17, 19	syl22anc 836	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦) ↔ (𝐹‘(𝐺‘𝑥))(le‘𝐾)(𝐹‘(𝐺‘𝑦))))
21	1, 18, 2	lautle 37222	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦)))
22	21	3adantl2 1163	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦)))
23		f1of 6617	. . . . . . 7 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
24	6, 23	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
25		simpl 485	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26		fvco3 6762	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
27	24, 25, 26	syl2an 597	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
28		simpr 487	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29		fvco3 6762	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
30	24, 28, 29	syl2an 597	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
31	27, 30	breq12d 5081	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (((𝐹 ∘ 𝐺)‘𝑥)(le‘𝐾)((𝐹 ∘ 𝐺)‘𝑦) ↔ (𝐹‘(𝐺‘𝑥))(le‘𝐾)(𝐹‘(𝐺‘𝑦))))
32	20, 22, 31	3bitr4d 313	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ ((𝐹 ∘ 𝐺)‘𝑥)(le‘𝐾)((𝐹 ∘ 𝐺)‘𝑦)))
33	32	ralrimivva 3193	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹 ∘ 𝐺)‘𝑥)(le‘𝐾)((𝐹 ∘ 𝐺)‘𝑦)))
34	1, 18, 2	islaut 37221	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((𝐹 ∘ 𝐺) ∈ 𝐼 ↔ ((𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹 ∘ 𝐺)‘𝑥)(le‘𝐾)((𝐹 ∘ 𝐺)‘𝑦)))))
35	34	3ad2ant1 1129	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → ((𝐹 ∘ 𝐺) ∈ 𝐼 ↔ ((𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ ((𝐹 ∘ 𝐺)‘𝑥)(le‘𝐾)((𝐹 ∘ 𝐺)‘𝑦)))))
36	8, 33, 35	mpbir2and 711	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺) ∈ 𝐼)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140   class class class wbr 5068   ∘ ccom 5561  ⟶wf 6353  –1-1-onto→wf1o 6356  ‘cfv 6357  Basecbs 16485  lecple 16574  LAutclaut 37123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-laut 37127

This theorem is referenced by:  ldilco  37254