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

Theorem motcgr 28291
Description: Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
ismot.p 𝑃 = (Baseβ€˜πΊ)
ismot.m βˆ’ = (distβ€˜πΊ)
motgrp.1 (πœ‘ β†’ 𝐺 ∈ 𝑉)
motcgr.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
motcgr.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
motcgr.f (πœ‘ β†’ 𝐹 ∈ (𝐺Ismt𝐺))
Assertion
Ref Expression
motcgr (πœ‘ β†’ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)) = (𝐴 βˆ’ 𝐡))

Proof of Theorem motcgr
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 motcgr.a . 2 (πœ‘ β†’ 𝐴 ∈ 𝑃)
2 motcgr.b . 2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
3 motcgr.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝐺Ismt𝐺))
4 motgrp.1 . . . . 5 (πœ‘ β†’ 𝐺 ∈ 𝑉)
5 ismot.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
6 ismot.m . . . . . 6 βˆ’ = (distβ€˜πΊ)
75, 6ismot 28290 . . . . 5 (𝐺 ∈ 𝑉 β†’ (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Ž βˆ’ 𝑏))))
84, 7syl 17 . . . 4 (πœ‘ β†’ (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Ž βˆ’ 𝑏))))
93, 8mpbid 231 . . 3 (πœ‘ β†’ (𝐹:𝑃–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Ž βˆ’ 𝑏)))
109simprd 495 . 2 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Ž βˆ’ 𝑏))
11 fveq2 6884 . . . . 5 (π‘Ž = 𝐴 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π΄))
1211oveq1d 7419 . . . 4 (π‘Ž = 𝐴 β†’ ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π‘)))
13 oveq1 7411 . . . 4 (π‘Ž = 𝐴 β†’ (π‘Ž βˆ’ 𝑏) = (𝐴 βˆ’ 𝑏))
1412, 13eqeq12d 2742 . . 3 (π‘Ž = 𝐴 β†’ (((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Ž βˆ’ 𝑏) ↔ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π‘)) = (𝐴 βˆ’ 𝑏)))
15 fveq2 6884 . . . . 5 (𝑏 = 𝐡 β†’ (πΉβ€˜π‘) = (πΉβ€˜π΅))
1615oveq2d 7420 . . . 4 (𝑏 = 𝐡 β†’ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π‘)) = ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)))
17 oveq2 7412 . . . 4 (𝑏 = 𝐡 β†’ (𝐴 βˆ’ 𝑏) = (𝐴 βˆ’ 𝐡))
1816, 17eqeq12d 2742 . . 3 (𝑏 = 𝐡 β†’ (((πΉβ€˜π΄) βˆ’ (πΉβ€˜π‘)) = (𝐴 βˆ’ 𝑏) ↔ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)) = (𝐴 βˆ’ 𝐡)))
1914, 18rspc2va 3618 . 2 (((𝐴 ∈ 𝑃 ∧ 𝐡 ∈ 𝑃) ∧ βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Ž βˆ’ 𝑏)) β†’ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)) = (𝐴 βˆ’ 𝐡))
201, 2, 10, 19syl21anc 835 1 (πœ‘ β†’ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)) = (𝐴 βˆ’ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  distcds 17213  Ismtcismt 28287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-ismt 28288
This theorem is referenced by:  motco  28295  cnvmot  28296  motcgrg  28299  motcgr3  28300
  Copyright terms: Public domain W3C validator