Theorem motco 28563

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 28561	. . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
6		motco.3	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺))
7	1, 2, 3, 6	motf1o 28561	. . 3 ⊢ (𝜑 → 𝐻:𝑃–1-1-onto→𝑃)
8		f1oco 6872	. . 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 584	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝑃–1-1-onto→𝑃)
10		f1of 6849	. . . . . . . 8 ⊢ (𝐻:𝑃–1-1-onto→𝑃 → 𝐻:𝑃⟶𝑃)
11	7, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑃⟶𝑃)
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐻:𝑃⟶𝑃)
13		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃)
14		fvco3 7008	. . . . . 6 ⊢ ((𝐻:𝑃⟶𝑃 ∧ 𝑎 ∈ 𝑃) → ((𝐹 ∘ 𝐻)‘𝑎) = (𝐹‘(𝐻‘𝑎)))
15	12, 13, 14	syl2anc 584	. . . . 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 7008	. . . . . 6 ⊢ ((𝐻:𝑃⟶𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹 ∘ 𝐻)‘𝑏) = (𝐹‘(𝐻‘𝑏)))
18	12, 16, 17	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹 ∘ 𝐻)‘𝑏) = (𝐹‘(𝐻‘𝑏)))
19	15, 18	oveq12d 7449	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = ((𝐹‘(𝐻‘𝑎)) − (𝐹‘(𝐻‘𝑏))))
20	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ 𝑉)
21	12, 13	ffvelcdmd 7105	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐻‘𝑎) ∈ 𝑃)
22	12, 16	ffvelcdmd 7105	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐻‘𝑏) ∈ 𝑃)
23	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
24	1, 2, 20, 21, 22, 23	motcgr 28559	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(𝐻‘𝑎)) − (𝐹‘(𝐻‘𝑏))) = ((𝐻‘𝑎) − (𝐻‘𝑏)))
25	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐻 ∈ (𝐺Ismt𝐺))
26	1, 2, 20, 13, 16, 25	motcgr 28559	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐻‘𝑎) − (𝐻‘𝑏)) = (𝑎 − 𝑏))
27	19, 24, 26	3eqtrd 2779	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = (𝑎 − 𝑏))
28	27	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝐹 ∘ 𝐻)‘𝑎) − ((𝐹 ∘ 𝐻)‘𝑏)) = (𝑎 − 𝑏))
29	1, 2	ismot 28558	. . 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∘ ccom 5693  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  Ismtcismt 28555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ismt 28556

This theorem is referenced by:  motgrp  28566