Theorem ldilco 35925

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 1239	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐾 ∈ 𝑉)
2		ldilco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2771	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ldilco.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
5	2, 3, 4	ldillaut 35920	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾))
6	5	3adant3 1126	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾))
7	2, 3, 4	ldillaut 35920	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ (LAut‘𝐾))
8	7	3adant2 1125	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ (LAut‘𝐾))
9	3	lautco 35906	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝐺 ∈ (LAut‘𝐾)) → (𝐹 ∘ 𝐺) ∈ (LAut‘𝐾))
10	1, 6, 8, 9	syl3anc 1476	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ (LAut‘𝐾))
11		simp11 1245	. . . . . . . 8 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
12		simp13 1247	. . . . . . . 8 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐺 ∈ 𝐷)
13		eqid 2771	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 2, 4	ldil1o 35921	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
15	11, 12, 14	syl2anc 573	. . . . . . 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 6279	. . . . . . 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 1131	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝑥 ∈ (Base‘𝐾))
19		fvco3 6419	. . . . . 6 ⊢ ((𝐺:(Base‘𝐾)⟶(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
20	17, 18, 19	syl2anc 573	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
21		simp3 1132	. . . . . . 7 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝑥(le‘𝐾)𝑊)
22		eqid 2771	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
23	13, 22, 2, 4	ldilval 35922	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝐷 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐺‘𝑥) = 𝑥)
24	11, 12, 18, 21, 23	syl112anc 1480	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐺‘𝑥) = 𝑥)
25	24	fveq2d 6337	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐹‘(𝐺‘𝑥)) = (𝐹‘𝑥))
26		simp12 1246	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → 𝐹 ∈ 𝐷)
27	13, 22, 2, 4	ldilval 35922	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐹‘𝑥) = 𝑥)
28	11, 26, 18, 21, 27	syl112anc 1480	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐹‘𝑥) = 𝑥)
29	20, 25, 28	3eqtrd 2809	. . . 4 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥)
30	29	3exp 1112	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝑥 ∈ (Base‘𝐾) → (𝑥(le‘𝐾)𝑊 → ((𝐹 ∘ 𝐺)‘𝑥) = 𝑥)))
31	30	ralrimiv 3114	. 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 35919	. . 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 1127	. 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 692	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘ 𝐺) ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061   class class class wbr 4787   ∘ ccom 5254  ⟶wf 6026  –1-1-onto→wf1o 6029  ‘cfv 6030  Basecbs 16064  lecple 16156  LHypclh 35793  LAutclaut 35794  LDilcldil 35909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-laut 35798  df-ldil 35913

This theorem is referenced by:  ltrnco  36529