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Theorem cnvmot 26439
 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 (𝜑𝐺𝑉)
motco.2 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
Assertion
Ref Expression
cnvmot (𝜑𝐹 ∈ (𝐺Ismt𝐺))

Proof of Theorem cnvmot
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismot.p . . . 4 𝑃 = (Base‘𝐺)
2 ismot.m . . . 4 = (dist‘𝐺)
3 motgrp.1 . . . 4 (𝜑𝐺𝑉)
4 motco.2 . . . 4 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
51, 2, 3, 4motf1o 26436 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 f1ocnv 6618 . . 3 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃1-1-onto𝑃)
75, 6syl 17 . 2 (𝜑𝐹:𝑃1-1-onto𝑃)
83adantr 484 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
9 f1of 6606 . . . . . . . 8 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃𝑃)
107, 9syl 17 . . . . . . 7 (𝜑𝐹:𝑃𝑃)
1110adantr 484 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃𝑃)
12 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
1311, 12ffvelrnd 6848 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑎) ∈ 𝑃)
14 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
1511, 14ffvelrnd 6848 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑏) ∈ 𝑃)
164adantr 484 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
171, 2, 8, 13, 15, 16motcgr 26434 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = ((𝐹𝑎) (𝐹𝑏)))
18 f1ocnvfv2 7031 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑎𝑃) → (𝐹‘(𝐹𝑎)) = 𝑎)
195, 12, 18syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑎)) = 𝑎)
20 f1ocnvfv2 7031 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑏𝑃) → (𝐹‘(𝐹𝑏)) = 𝑏)
215, 14, 20syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑏)) = 𝑏)
2219, 21oveq12d 7173 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
2317, 22eqtr3d 2795 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
2423ralrimivva 3120 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
251, 2ismot 26433 . . 3 (𝐺𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
263, 25syl 17 . 2 (𝜑 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
277, 24, 26mpbir2and 712 1 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ◡ccnv 5526  ⟶wf 6335  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  distcds 16637  Ismtcismt 26430 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 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8423  df-ismt 26431 This theorem is referenced by:  motgrp  26441
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