Theorem cnvmot 26335

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 26332	. . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
6		f1ocnv 6602	. . 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 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ 𝑉)
9		f1of 6590	. . . . . . . 8 ⊢ (◡𝐹:𝑃–1-1-onto→𝑃 → ◡𝐹:𝑃⟶𝑃)
10	7, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → ◡𝐹:𝑃⟶𝑃)
11	10	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ◡𝐹:𝑃⟶𝑃)
12		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃)
13	11, 12	ffvelrnd 6829	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑎) ∈ 𝑃)
14		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃)
15	11, 14	ffvelrnd 6829	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑏) ∈ 𝑃)
16	4	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
17	1, 2, 8, 13, 15, 16	motcgr 26330	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)))
18		f1ocnvfv2 7012	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑎 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
19	5, 12, 18	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
20		f1ocnvfv2 7012	. . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑏 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
21	5, 14, 20	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
22	19, 21	oveq12d 7153	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = (𝑎 − 𝑏))
23	17, 22	eqtr3d 2835	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
24	23	ralrimivva 3156	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏))
25	1, 2	ismot 26329	. . 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 712	1 ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ^◡ccnv 5518  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  distcds 16566  Ismtcismt 26326

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-ismt 26327

This theorem is referenced by:  motgrp  26337