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

Theorem idmot 26015
Description: The identity 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)
Assertion
Ref Expression
idmot (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺))

Proof of Theorem idmot
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 motgrp.1 . 2 (𝜑𝐺𝑉)
2 f1oi 6475 . . 3 ( I ↾ 𝑃):𝑃1-1-onto𝑃
32a1i 11 . 2 (𝜑 → ( I ↾ 𝑃):𝑃1-1-onto𝑃)
4 fvresi 6752 . . . . 5 (𝑎𝑃 → (( I ↾ 𝑃)‘𝑎) = 𝑎)
54ad2antrl 715 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (( I ↾ 𝑃)‘𝑎) = 𝑎)
6 fvresi 6752 . . . . 5 (𝑏𝑃 → (( I ↾ 𝑃)‘𝑏) = 𝑏)
76ad2antll 716 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (( I ↾ 𝑃)‘𝑏) = 𝑏)
85, 7oveq12d 6988 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((( I ↾ 𝑃)‘𝑎) (( I ↾ 𝑃)‘𝑏)) = (𝑎 𝑏))
98ralrimivva 3135 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 ((( I ↾ 𝑃)‘𝑎) (( I ↾ 𝑃)‘𝑏)) = (𝑎 𝑏))
10 ismot.p . . . 4 𝑃 = (Base‘𝐺)
11 ismot.m . . . 4 = (dist‘𝐺)
1210, 11ismot 26013 . . 3 (𝐺𝑉 → (( I ↾ 𝑃) ∈ (𝐺Ismt𝐺) ↔ (( I ↾ 𝑃):𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((( I ↾ 𝑃)‘𝑎) (( I ↾ 𝑃)‘𝑏)) = (𝑎 𝑏))))
1312biimpar 470 . 2 ((𝐺𝑉 ∧ (( I ↾ 𝑃):𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((( I ↾ 𝑃)‘𝑎) (( I ↾ 𝑃)‘𝑏)) = (𝑎 𝑏))) → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺))
141, 3, 9, 13syl12anc 824 1 (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  wral 3082   I cid 5304  cres 5402  1-1-ontowf1o 6181  cfv 6182  (class class class)co 6970  Basecbs 16329  distcds 16420  Ismtcismt 26010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-map 8200  df-ismt 26011
This theorem is referenced by:  motgrp  26021
  Copyright terms: Public domain W3C validator