Theorem cnvmot 28549

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 28546	. . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
6		f1ocnv 6860	. . 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 6848	. . . . . . . 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 7105	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑎) ∈ 𝑃)
14		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃)
15	11, 14	ffvelcdmd 7105	. . . . 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 28544	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)))
18		f1ocnvfv2 7297	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑎 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
19	5, 12, 18	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
20		f1ocnvfv2 7297	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑏 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
21	5, 14, 20	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
22	19, 21	oveq12d 7449	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = (𝑎 − 𝑏))
23	17, 22	eqtr3d 2779	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
24	23	ralrimivva 3202	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
25	1, 2	ismot 28543	. . 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 713	1 ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ^◡ccnv 5684  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  Ismtcismt 28540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-ismt 28541

This theorem is referenced by:  motgrp  28551