Theorem motco 26631

Description: The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
ismot.p	⊢ 𝑃 = (Base‘𝐺)
ismot.m	⊢ − = (dist‘𝐺)
motgrp.1	⊢ (𝜑 → 𝐺 ∈ 𝑉)
motco.2	⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺))
motco.3	⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺))

Assertion

Ref	Expression
motco	⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐺Ismt𝐺))

Proof of Theorem motco

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismot.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		ismot.m	. . . 4 ⊢ − = (dist‘𝐺)
3		motgrp.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
4		motco.2	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺))
5	1, 2, 3, 4	motf1o 26629	. . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
6		motco.3	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺))
7	1, 2, 3, 6	motf1o 26629	. . 3 ⊢ (𝜑 → 𝐻:𝑃–1-1-onto→𝑃)
8		f1oco 6683	. . 3 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝐻:𝑃–1-1-onto→𝑃) → (𝐹 ∘ 𝐻):𝑃–1-1-onto→𝑃)
9	5, 7, 8	syl2anc 587	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝑃–1-1-onto→𝑃)
10		f1of 6661	. . . . . . . 8 ⊢ (𝐻:𝑃–1-1-onto→𝑃 → 𝐻:𝑃⟶𝑃)
11	7, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑃⟶𝑃)
12	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐻:𝑃⟶𝑃)
13		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃)
14		fvco3 6810	. . . . . 6 ⊢ ((𝐻:𝑃⟶𝑃 ∧ 𝑎 ∈ 𝑃) → ((𝐹 ∘ 𝐻)‘𝑎) = (𝐹‘(𝐻‘𝑎)))
15	12, 13, 14	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹 ∘ 𝐻)‘𝑎) = (𝐹‘(𝐻‘𝑎)))
16		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃)
17		fvco3 6810	. . . . . 6 ⊢ ((𝐻:𝑃⟶𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹 ∘ 𝐻)‘𝑏) = (𝐹‘(𝐻‘𝑏)))
18	12, 16, 17	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹 ∘ 𝐻)‘𝑏) = (𝐹‘(𝐻‘𝑏)))
19	15, 18	oveq12d 7231	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = ((𝐹‘(𝐻‘𝑎)) − (𝐹‘(𝐻‘𝑏))))
20	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ 𝑉)
21	12, 13	ffvelrnd 6905	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐻‘𝑎) ∈ 𝑃)
22	12, 16	ffvelrnd 6905	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐻‘𝑏) ∈ 𝑃)
23	4	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
24	1, 2, 20, 21, 22, 23	motcgr 26627	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(𝐻‘𝑎)) − (𝐹‘(𝐻‘𝑏))) = ((𝐻‘𝑎) − (𝐻‘𝑏)))
25	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐻 ∈ (𝐺Ismt𝐺))
26	1, 2, 20, 13, 16, 25	motcgr 26627	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐻‘𝑎) − (𝐻‘𝑏)) = (𝑎 − 𝑏))
27	19, 24, 26	3eqtrd 2781	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = (𝑎 − 𝑏))
28	27	ralrimivva 3112	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = (𝑎 − 𝑏))
29	1, 2	ismot 26626	. . 3 ⊢ (𝐺 ∈ 𝑉 → ((𝐹 ∘ 𝐻) ∈ (𝐺Ismt𝐺) ↔ ((𝐹 ∘ 𝐻):𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = (𝑎 − 𝑏))))
30	3, 29	syl 17	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∈ (𝐺Ismt𝐺) ↔ ((𝐹 ∘ 𝐻):𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = (𝑎 − 𝑏))))
31	9, 28, 30	mpbir2and 713	1 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐺Ismt𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061   ∘ ccom 5555  ⟶wf 6376  –1-1-onto→wf1o 6379  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  distcds 16811  Ismtcismt 26623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-ismt 26624

This theorem is referenced by:  motgrp  26634