Theorem lautco 38560

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 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautco.i	. . . . 5 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 38548	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4	3	3adant3 1132	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	1, 2	laut1o 38548	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1131	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1oco 6807	. . 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 584	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		simpl1 1191	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾 ∈ 𝑉)
10		simpl2 1192	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹 ∈ 𝐼)
11		simpl3 1193	. . . . . 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 38550	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
14	9, 11, 12, 13	syl21anc 836	. . . . 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 38550	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺‘𝑦) ∈ (Base‘𝐾))
17	9, 11, 15, 16	syl21anc 836	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑦) ∈ (Base‘𝐾))
18		eqid 2736	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
19	1, 18, 2	lautle 38547	. . . . 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 837	. . . 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 38547	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦)))
22	21	3adantl2 1167	. . . 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 6784	. . . . . . 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 483	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26		fvco3 6940	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
27	24, 25, 26	syl2an 596	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
28		simpr 485	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29		fvco3 6940	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
30	24, 28, 29	syl2an 596	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
31	27, 30	breq12d 5118	. . . 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 310	. . 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 3197	. 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 38546	. . 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 1133	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   class class class wbr 5105   ∘ ccom 5637  ⟶wf 6492  –1-1-onto→wf1o 6495  ‘cfv 6496  Basecbs 17083  lecple 17140  LAutclaut 38448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-laut 38452

This theorem is referenced by:  ldilco  38579