Theorem cnvmot 28609

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 28606	. . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
6		f1ocnv 6792	. . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ 𝑉)
9		f1of 6780	. . . . . . . 8 ⊢ (◡𝐹:𝑃–1-1-onto→𝑃 → ◡𝐹:𝑃⟶𝑃)
10	7, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → ◡𝐹:𝑃⟶𝑃)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ◡𝐹:𝑃⟶𝑃)
12		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃)
13	11, 12	ffvelcdmd 7037	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑎) ∈ 𝑃)
14		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃)
15	11, 14	ffvelcdmd 7037	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑏) ∈ 𝑃)
16	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
17	1, 2, 8, 13, 15, 16	motcgr 28604	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)))
18		f1ocnvfv2 7232	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑎 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
19	5, 12, 18	syl2an2r 686	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
20		f1ocnvfv2 7232	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑏 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
21	5, 14, 20	syl2an2r 686	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
22	19, 21	oveq12d 7385	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = (𝑎 − 𝑏))
23	17, 22	eqtr3d 2773	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
24	23	ralrimivva 3180	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
25	1, 2	ismot 28603	. . 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 714	1 ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ^◡ccnv 5630  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  distcds 17229  Ismtcismt 28600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-ismt 28601

This theorem is referenced by:  motgrp  28611