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Theorem motcgr 28353
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 28352 . . . . 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 6897 . . . . 5 (π‘Ž = 𝐴 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π΄))
1211oveq1d 7435 . . . 4 (π‘Ž = 𝐴 β†’ ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π‘)))
13 oveq1 7427 . . . 4 (π‘Ž = 𝐴 β†’ (π‘Ž βˆ’ 𝑏) = (𝐴 βˆ’ 𝑏))
1412, 13eqeq12d 2744 . . 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 6897 . . . . 5 (𝑏 = 𝐡 β†’ (πΉβ€˜π‘) = (πΉβ€˜π΅))
1615oveq2d 7436 . . . 4 (𝑏 = 𝐡 β†’ ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π‘)) = ((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)))
17 oveq2 7428 . . . 4 (𝑏 = 𝐡 β†’ (𝐴 βˆ’ 𝑏) = (𝐴 βˆ’ 𝐡))
1816, 17eqeq12d 2744 . . 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 3621 . 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 837 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 1534   ∈ wcel 2099  βˆ€wral 3058  β€“1-1-ontoβ†’wf1o 6547  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  distcds 17242  Ismtcismt 28349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-ismt 28350
This theorem is referenced by:  motco  28357  cnvmot  28358  motcgrg  28361  motcgr3  28362
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