Theorem ldilco 40140

Description: The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

Hypotheses

Ref	Expression
ldilco.h	⊢ 𝐻 = (LHyp‘𝐾)
ldilco.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)

Assertion

Ref	Expression
ldilco	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ 𝐷)

Proof of Theorem ldilco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1198	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐾 ∈ 𝑉)
2		ldilco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2736	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ldilco.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
5	2, 3, 4	ldillaut 40135	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾))
6	5	3adant3 1132	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾))
7	2, 3, 4	ldillaut 40135	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ (LAut‘𝐾))
8	7	3adant2 1131	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ (LAut‘𝐾))
9	3	lautco 40121	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝐺 ∈ (LAut‘𝐾)) → (𝐹 ∘ 𝐺) ∈ (LAut‘𝐾))
10	1, 6, 8, 9	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ (LAut‘𝐾))
11		simp11 1204	. . . . . . . 8 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
12		simp13 1206	. . . . . . . 8 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐺 ∈ 𝐷)
13		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 2, 4	ldil1o 40136	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
15	11, 12, 14	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
16		f1of 6823	. . . . . . 7 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
17	15, 16	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
18		simp2 1137	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝑥 ∈ (Base‘𝐾))
19		fvco3 6983	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
20	17, 18, 19	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
21		simp3 1138	. . . . . . 7 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝑥(le‘𝐾)𝑊)
22		eqid 2736	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
23	13, 22, 2, 4	ldilval 40137	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐺‘𝑥) = 𝑥)
24	11, 12, 18, 21, 23	syl112anc 1376	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐺‘𝑥) = 𝑥)
25	24	fveq2d 6885	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐹‘(𝐺‘𝑥)) = (𝐹‘𝑥))
26		simp12 1205	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐹 ∈ 𝐷)
27	13, 22, 2, 4	ldilval 40137	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐹‘𝑥) = 𝑥)
28	11, 26, 18, 21, 27	syl112anc 1376	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐹‘𝑥) = 𝑥)
29	20, 25, 28	3eqtrd 2775	. . . 4 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥)
30	29	3exp 1119	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝑥 ∈ (Base‘𝐾) → (𝑥(le‘𝐾)𝑊 → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥)))
31	30	ralrimiv 3132	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑊 → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥))
32	13, 22, 2, 3, 4	isldil 40134	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((𝐹 ∘ 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘ 𝐺) ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑊 → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥))))
33	32	3ad2ant1 1133	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘ 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘ 𝐺) ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑊 → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥))))
34	10, 31, 33	mpbir2and 713	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   class class class wbr 5124   ∘ ccom 5663  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  Basecbs 17233  lecple 17283  LHypclh 40008  LAutclaut 40009  LDilcldil 40124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-laut 40013  df-ldil 40128

This theorem is referenced by:  ltrnco  40743