Theorem cnvmot 28627

Description: The converse of a motion 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))

Assertion

Ref	Expression
cnvmot	⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺))

Proof of Theorem cnvmot

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 28624	. . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
6		f1ocnv 6779	. . 3 ⊢ (𝐹:𝑃–1-1-onto→𝑃 → ◡𝐹:𝑃–1-1-onto→𝑃)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ◡𝐹:𝑃–1-1-onto→𝑃)
8	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ 𝑉)
9		f1of 6767	. . . . . . . 8 ⊢ (◡𝐹:𝑃–1-1-onto→𝑃 → ◡𝐹:𝑃⟶𝑃)
10	7, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → ◡𝐹:𝑃⟶𝑃)
11	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ◡𝐹:𝑃⟶𝑃)
12		simprl 776	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃)
13	11, 12	ffvelcdmd 7026	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑎) ∈ 𝑃)
14		simprr 778	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃)
15	11, 14	ffvelcdmd 7026	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑏) ∈ 𝑃)
16	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
17	1, 2, 8, 13, 15, 16	motcgr 28622	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)))
18		f1ocnvfv2 7221	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑎 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
19	5, 12, 18	syl2an2r 691	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
20		f1ocnvfv2 7221	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑏 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
21	5, 14, 20	syl2an2r 691	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
22	19, 21	oveq12d 7374	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = (𝑎 − 𝑏))
23	17, 22	eqtr3d 2776	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
24	23	ralrimivva 3182	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
25	1, 2	ismot 28621	. . 3 ⊢ (𝐺 ∈ 𝑉 → (◡𝐹 ∈ (𝐺Ismt𝐺) ↔ (◡𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))))
26	3, 25	syl 17	. 2 ⊢ (𝜑 → (◡𝐹 ∈ (𝐺Ismt𝐺) ↔ (◡𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))))
27	7, 24, 26	mpbir2and 719	1 ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ^◡ccnv 5617  ⟶wf 6481  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  distcds 17220  Ismtcismt 28618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-ismt 28619

This theorem is referenced by:  motgrp  28629