Theorem ldilco 34420

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 1078	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐾 ∈ 𝑉)
2		ldilco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2610	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ldilco.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
5	2, 3, 4	ldillaut 34415	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾))
6	5	3adant3 1074	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾))
7	2, 3, 4	ldillaut 34415	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ (LAut‘𝐾))
8	7	3adant2 1073	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ (LAut‘𝐾))
9	3	lautco 34401	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝐺 ∈ (LAut‘𝐾)) → (𝐹 ∘ 𝐺) ∈ (LAut‘𝐾))
10	1, 6, 8, 9	syl3anc 1318	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ (LAut‘𝐾))
11		simp11 1084	. . . . . . . 8 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
12		simp13 1086	. . . . . . . 8 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐺 ∈ 𝐷)
13		eqid 2610	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 2, 4	ldil1o 34416	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
15	11, 12, 14	syl2anc 691	. . . . . . 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 6050	. . . . . . 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 1055	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝑥 ∈ (Base‘𝐾))
19		fvco3 6185	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
20	17, 18, 19	syl2anc 691	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
21		simp3 1056	. . . . . . 7 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝑥(le‘𝐾)𝑊)
22		eqid 2610	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
23	13, 22, 2, 4	ldilval 34417	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐺‘𝑥) = 𝑥)
24	11, 12, 18, 21, 23	syl112anc 1322	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐺‘𝑥) = 𝑥)
25	24	fveq2d 6107	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐹‘(𝐺‘𝑥)) = (𝐹‘𝑥))
26		simp12 1085	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐹 ∈ 𝐷)
27	13, 22, 2, 4	ldilval 34417	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐹‘𝑥) = 𝑥)
28	11, 26, 18, 21, 27	syl112anc 1322	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐹‘𝑥) = 𝑥)
29	20, 25, 28	3eqtrd 2648	. . . 4 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥)
30	29	3exp 1256	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝑥 ∈ (Base‘𝐾) → (𝑥(le‘𝐾)𝑊 → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥)))
31	30	ralrimiv 2948	. 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 34414	. . 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 1075	. 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 959	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LAutclaut 34289  LDilcldil 34404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-laut 34293  df-ldil 34408

This theorem is referenced by:  ltrnco  35025