Theorem motcgr 25431

Description: Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
ismot.p	|- P = ( Base ` G )
ismot.m	|- .- = ( dist ` G )
motgrp.1	|- ( ph -> G e. V )
motcgr.a	|- ( ph -> A e. P )
motcgr.b	|- ( ph -> B e. P )
motcgr.f	|- ( ph -> F e. ( GIsmt G ) )

Assertion

Ref	Expression
motcgr	|- ( ph -> ( ( F ` A ) .- ( F ` B ) ) = ( A .- B ) )

Proof of Theorem motcgr

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		motcgr.a	. 2 |- ( ph -> A e. P )
2		motcgr.b	. 2 |- ( ph -> B e. P )
3		motcgr.f	. . . 4 |- ( ph -> F e. ( GIsmt G ) )
4		motgrp.1	. . . . 5 |- ( ph -> G e. V )
5		ismot.p	. . . . . 6 |- P = ( Base ` G )
6		ismot.m	. . . . . 6 |- .- = ( dist ` G )
7	5, 6	ismot 25430	. . . . 5 |- ( G e. V -> ( F e. ( GIsmt G ) <-> ( F : P -1-1-onto-> P /\ A. a e. P A. b e. P ( ( F ` a ) .- ( F ` b ) ) = ( a .- b ) ) ) )
8	4, 7	syl 17	. . . 4 |- ( ph -> ( F e. ( GIsmt G ) <-> ( F : P -1-1-onto-> P /\ A. a e. P A. b e. P ( ( F ` a ) .- ( F ` b ) ) = ( a .- b ) ) ) )
9	3, 8	mpbid 222	. . 3 |- ( ph -> ( F : P -1-1-onto-> P /\ A. a e. P A. b e. P ( ( F ` a ) .- ( F ` b ) ) = ( a .- b ) ) )
10	9	simprd 479	. 2 |- ( ph -> A. a e. P A. b e. P ( ( F ` a ) .- ( F ` b ) ) = ( a .- b ) )
11		fveq2 6191	. . . . 5 |- ( a = A -> ( F ` a ) = ( F ` A ) )
12	11	oveq1d 6665	. . . 4 |- ( a = A -> ( ( F ` a ) .- ( F ` b ) ) = ( ( F ` A ) .- ( F ` b ) ) )
13		oveq1 6657	. . . 4 |- ( a = A -> ( a .- b ) = ( A .- b ) )
14	12, 13	eqeq12d 2637	. . 3 |- ( a = A -> ( ( ( F ` a ) .- ( F ` b ) ) = ( a .- b ) <-> ( ( F ` A ) .- ( F ` b ) ) = ( A .- b ) ) )
15		fveq2 6191	. . . . 5 |- ( b = B -> ( F ` b ) = ( F ` B ) )
16	15	oveq2d 6666	. . . 4 |- ( b = B -> ( ( F ` A ) .- ( F ` b ) ) = ( ( F ` A ) .- ( F ` B ) ) )
17		oveq2 6658	. . . 4 |- ( b = B -> ( A .- b ) = ( A .- B ) )
18	16, 17	eqeq12d 2637	. . 3 |- ( b = B -> ( ( ( F ` A ) .- ( F ` b ) ) = ( A .- b ) <-> ( ( F ` A ) .- ( F ` B ) ) = ( A .- B ) ) )
19	14, 18	rspc2va 3323	. 2 |- ( ( ( A e. P /\ B e. P ) /\ A. a e. P A. b e. P ( ( F ` a ) .- ( F ` b ) ) = ( a .- b ) ) -> ( ( F ` A ) .- ( F ` B ) ) = ( A .- B ) )
20	1, 2, 10, 19	syl21anc 1325	1 |- ( ph -> ( ( F ` A ) .- ( F ` B ) ) = ( A .- B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  Ismtcismt 25427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ismt 25428

This theorem is referenced by:  motco  25435  cnvmot  25436  motcgrg  25439  motcgr3  25440