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	⊢ ((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 2778	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautco.i	. . . . 5 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 36248	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4	3	3adant3 1123	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	1, 2	laut1o 36248	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1122	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 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)) → (𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8	4, 6, 7	syl2anc 579	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		simpl1 1199	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾 ∈ 𝑉)
10		simpl2 1201	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹 ∈ 𝐼)
11		simpl3 1203	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺 ∈ 𝐼)
12		simprl 761	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
13	1, 2	lautcl 36250	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
14	9, 11, 12, 13	syl21anc 828	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑥) ∈ (Base‘𝐾))
15		simprr 763	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
16	1, 2	lautcl 36250	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺‘𝑦) ∈ (Base‘𝐾))
17	9, 11, 15, 16	syl21anc 828	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑦) ∈ (Base‘𝐾))
18		eqid 2778	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
19	1, 18, 2	lautle 36247	. . . . 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 829	. . . 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 36247	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦)))
22	21	3adantl2 1169	. . . 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 6393	. . . . . . 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 476	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26		fvco3 6537	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
27	24, 25, 26	syl2an 589	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
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)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
30	24, 28, 29	syl2an 589	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
31	27, 30	breq12d 4901	. . . 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 303	. . 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 3153	. 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 36246	. . 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 1124	. 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 703	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺) ∈ 𝐼)

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-onto→wf1o 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