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

Theorem cnvmot 25792
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 25789 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 f1ocnv 6368 . . 3 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃1-1-onto𝑃)
75, 6syl 17 . 2 (𝜑𝐹:𝑃1-1-onto𝑃)
83adantr 473 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
9 f1of 6356 . . . . . . . 8 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃𝑃)
107, 9syl 17 . . . . . . 7 (𝜑𝐹:𝑃𝑃)
1110adantr 473 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃𝑃)
12 simprl 788 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
1311, 12ffvelrnd 6586 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑎) ∈ 𝑃)
14 simprr 790 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
1511, 14ffvelrnd 6586 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑏) ∈ 𝑃)
164adantr 473 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
171, 2, 8, 13, 15, 16motcgr 25787 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = ((𝐹𝑎) (𝐹𝑏)))
185adantr 473 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃1-1-onto𝑃)
19 f1ocnvfv2 6761 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑎𝑃) → (𝐹‘(𝐹𝑎)) = 𝑎)
2018, 12, 19syl2anc 580 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑎)) = 𝑎)
21 f1ocnvfv2 6761 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑏𝑃) → (𝐹‘(𝐹𝑏)) = 𝑏)
2218, 14, 21syl2anc 580 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑏)) = 𝑏)
2320, 22oveq12d 6896 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
2417, 23eqtr3d 2835 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
2524ralrimivva 3152 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
261, 2ismot 25786 . . 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 705 1 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  ccnv 5311  wf 6097  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  Basecbs 16184  distcds 16276  Ismtcismt 25783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-ismt 25784
This theorem is referenced by:  motgrp  25794
  Copyright terms: Public domain W3C validator