MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvmot Structured version   Visualization version   GIF version

Theorem cnvmot 25370
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 25367 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 f1ocnv 6116 . . 3 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃1-1-onto𝑃)
75, 6syl 17 . 2 (𝜑𝐹:𝑃1-1-onto𝑃)
83adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
9 f1of 6104 . . . . . . . 8 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃𝑃)
107, 9syl 17 . . . . . . 7 (𝜑𝐹:𝑃𝑃)
1110adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃𝑃)
12 simprl 793 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
1311, 12ffvelrnd 6326 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑎) ∈ 𝑃)
14 simprr 795 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
1511, 14ffvelrnd 6326 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑏) ∈ 𝑃)
164adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
171, 2, 8, 13, 15, 16motcgr 25365 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = ((𝐹𝑎) (𝐹𝑏)))
185adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃1-1-onto𝑃)
19 f1ocnvfv2 6498 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑎𝑃) → (𝐹‘(𝐹𝑎)) = 𝑎)
2018, 12, 19syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑎)) = 𝑎)
21 f1ocnvfv2 6498 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑏𝑃) → (𝐹‘(𝐹𝑏)) = 𝑏)
2218, 14, 21syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑏)) = 𝑏)
2320, 22oveq12d 6633 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
2417, 23eqtr3d 2657 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
2524ralrimivva 2967 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
261, 2ismot 25364 . . 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 956 1 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  ccnv 5083  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  Basecbs 15800  distcds 15890  Ismtcismt 25361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-ismt 25362
This theorem is referenced by:  motgrp  25372
  Copyright terms: Public domain W3C validator