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Theorem cnvmot 25656
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 25653 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 f1ocnv 6290 . . 3 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃1-1-onto𝑃)
75, 6syl 17 . 2 (𝜑𝐹:𝑃1-1-onto𝑃)
83adantr 466 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
9 f1of 6278 . . . . . . . 8 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃𝑃)
107, 9syl 17 . . . . . . 7 (𝜑𝐹:𝑃𝑃)
1110adantr 466 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃𝑃)
12 simprl 746 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
1311, 12ffvelrnd 6503 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑎) ∈ 𝑃)
14 simprr 748 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
1511, 14ffvelrnd 6503 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑏) ∈ 𝑃)
164adantr 466 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
171, 2, 8, 13, 15, 16motcgr 25651 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = ((𝐹𝑎) (𝐹𝑏)))
185adantr 466 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃1-1-onto𝑃)
19 f1ocnvfv2 6675 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑎𝑃) → (𝐹‘(𝐹𝑎)) = 𝑎)
2018, 12, 19syl2anc 565 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑎)) = 𝑎)
21 f1ocnvfv2 6675 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑏𝑃) → (𝐹‘(𝐹𝑏)) = 𝑏)
2218, 14, 21syl2anc 565 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑏)) = 𝑏)
2320, 22oveq12d 6810 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
2417, 23eqtr3d 2806 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
2524ralrimivva 3119 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
261, 2ismot 25650 . . 3 (𝐺𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
273, 26syl 17 . 2 (𝜑 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
287, 25, 27mpbir2and 684 1 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  ccnv 5248  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  Basecbs 16063  distcds 16157  Ismtcismt 25647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-ismt 25648
This theorem is referenced by:  motgrp  25658
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