Theorem cnvmot 25436

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	|- ( ph -> G e. V )
motco.2	|- ( ph -> F e. ( GIsmt G ) )

Assertion

Ref	Expression
cnvmot	|- ( ph -> `' F e. ( GIsmt G ) )

Proof of Theorem cnvmot

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

Step	Hyp	Ref	Expression
1		ismot.p	. . . 4 |- P = ( Base ` G )
2		ismot.m	. . . 4 |- .- = ( dist ` G )
3		motgrp.1	. . . 4 |- ( ph -> G e. V )
4		motco.2	. . . 4 |- ( ph -> F e. ( GIsmt G ) )
5	1, 2, 3, 4	motf1o 25433	. . 3 |- ( ph -> F : P -1-1-onto-> P )
6		f1ocnv 6149	. . 3 |- ( F : P -1-1-onto-> P -> `' F : P -1-1-onto-> P )
7	5, 6	syl 17	. 2 |- ( ph -> `' F : P -1-1-onto-> P )
8	3	adantr 481	. . . . 5 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> G e. V )
9		f1of 6137	. . . . . . . 8 |- ( `' F : P -1-1-onto-> P -> `' F : P --> P )
10	7, 9	syl 17	. . . . . . 7 |- ( ph -> `' F : P --> P )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> `' F : P --> P )
12		simprl 794	. . . . . 6 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> a e. P )
13	11, 12	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( `' F ` a ) e. P )
14		simprr 796	. . . . . 6 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> b e. P )
15	11, 14	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( `' F ` b ) e. P )
16	4	adantr 481	. . . . 5 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> F e. ( GIsmt G ) )
17	1, 2, 8, 13, 15, 16	motcgr 25431	. . . 4 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( ( F ` ( `' F ` a ) ) .- ( F ` ( `' F ` b ) ) ) = ( ( `' F ` a ) .- ( `' F ` b ) ) )
18	5	adantr 481	. . . . . 6 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> F : P -1-1-onto-> P )
19		f1ocnvfv2 6533	. . . . . 6 |- ( ( F : P -1-1-onto-> P /\ a e. P ) -> ( F ` ( `' F ` a ) ) = a )
20	18, 12, 19	syl2anc 693	. . . . 5 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( F ` ( `' F ` a ) ) = a )
21		f1ocnvfv2 6533	. . . . . 6 |- ( ( F : P -1-1-onto-> P /\ b e. P ) -> ( F ` ( `' F ` b ) ) = b )
22	18, 14, 21	syl2anc 693	. . . . 5 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( F ` ( `' F ` b ) ) = b )
23	20, 22	oveq12d 6668	. . . 4 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( ( F ` ( `' F ` a ) ) .- ( F ` ( `' F ` b ) ) ) = ( a .- b ) )
24	17, 23	eqtr3d 2658	. . 3 |- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( ( `' F ` a ) .- ( `' F ` b ) ) = ( a .- b ) )
25	24	ralrimivva 2971	. 2 |- ( ph -> A. a e. P A. b e. P ( ( `' F ` a ) .- ( `' F ` b ) ) = ( a .- b ) )
26	1, 2	ismot 25430	. . 3 |- ( 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 ) ) ) )
27	3, 26	syl 17	. 2 |- ( 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 ) ) ) )
28	7, 25, 27	mpbir2and 957	1 |- ( ph -> `' F e. ( GIsmt G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   `'ccnv 5113   -->wf 5884   -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:  motgrp  25438