Theorem lautco 41154

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 2761	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautco.i	. . . . 5 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 41142	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4	3	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	1, 2	laut1o 41142	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1149	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1oco 6848	. . 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 596	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		simpl1 1210	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐾 ∈ 𝑉)
10		simpl2 1211	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐹 ∈ 𝐼)
11		simpl3 1212	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐺 ∈ 𝐼)
12		simprl 783	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ (Base‘𝐾))
13	1, 2	lautcl 41144	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
14	9, 11, 12, 13	syl21anc 851	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑥) ∈ (Base‘𝐾))
15		simprr 785	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ (Base‘𝐾))
16	1, 2	lautcl 41144	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝐺‘𝑦) ∈ (Base‘𝐾))
17	9, 11, 15, 16	syl21anc 851	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝐺‘𝑦) ∈ (Base‘𝐾))
18		eqid 2761	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
19	1, 18, 2	lautle 41141	. . . . 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 852	. . . 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 41141	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑥)(le‘𝐾)(𝐺‘𝑦)))
22	21	3adantl2 1186	. . . 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 6824	. . . . . . 7 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
24	6, 23	syl 18	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
25		simpl 488	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
26		fvco3 6985	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
27	24, 25, 26	syl2an 608	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
28		simpr 490	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
29		fvco3 6985	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
30	24, 28, 29	syl2an 608	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
31	27, 30	breq12d 5116	. . . 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 314	. . 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 3206	. 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 41140	. . 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 1151	. 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 726	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼) → (𝐹 ∘ 𝐺) ∈ 𝐼)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103   ∘ ccom 5655  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  Basecbs 17387  lecple 17435  LAutclaut 41042

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-laut 41046

This theorem is used by:  ldilco  41173