Theorem motgrp 25438

Description: The motions of a geometry form a group with respect to function composition, called the Isometry group. (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 )
motgrp.i	|- I = { <. ( Base ` ndx ) , ( GIsmt G ) >. , <. ( +g ` ndx ) , ( f e. ( GIsmt G ) , g e. ( GIsmt G ) |-> ( f o. g ) ) >. }

Assertion

Ref	Expression
motgrp	|- ( ph -> I e. Grp )

Distinct variable groups:   f, G, g   f, I, g   P, f, g   ph, f, g

Allowed substitution hints:   .- ( f, g)   V( f, g)

Proof of Theorem motgrp

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

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 |- ( GIsmt G ) e. _V
2		motgrp.i	. . . 4 |- I = { <. ( Base ` ndx ) , ( GIsmt G ) >. , <. ( +g ` ndx ) , ( f e. ( GIsmt G ) , g e. ( GIsmt G ) |-> ( f o. g ) ) >. }
3	2	grpbase 15991	. . 3 |- ( ( GIsmt G ) e. _V -> ( GIsmt G ) = ( Base ` I ) )
4	1, 3	mp1i 13	. 2 |- ( ph -> ( GIsmt G ) = ( Base ` I ) )
5		eqidd 2623	. 2 |- ( ph -> ( +g ` I ) = ( +g ` I ) )
6		ismot.p	. . . 4 |- P = ( Base ` G )
7		ismot.m	. . . 4 |- .- = ( dist ` G )
8		motgrp.1	. . . . 5 |- ( ph -> G e. V )
9	8	3ad2ant1 1082	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) /\ g e. ( GIsmt G ) ) -> G e. V )
10		simp2 1062	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) /\ g e. ( GIsmt G ) ) -> f e. ( GIsmt G ) )
11		simp3 1063	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) /\ g e. ( GIsmt G ) ) -> g e. ( GIsmt G ) )
12	6, 7, 9, 2, 10, 11	motplusg 25437	. . 3 |- ( ( ph /\ f e. ( GIsmt G ) /\ g e. ( GIsmt G ) ) -> ( f ( +g ` I ) g ) = ( f o. g ) )
13	6, 7, 9, 10, 11	motco 25435	. . 3 |- ( ( ph /\ f e. ( GIsmt G ) /\ g e. ( GIsmt G ) ) -> ( f o. g ) e. ( GIsmt G ) )
14	12, 13	eqeltrd 2701	. 2 |- ( ( ph /\ f e. ( GIsmt G ) /\ g e. ( GIsmt G ) ) -> ( f ( +g ` I ) g ) e. ( GIsmt G ) )
15		coass 5654	. . 3 |- ( ( f o. g ) o. h ) = ( f o. ( g o. h ) )
16	12	3adant3r3 1276	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( f ( +g ` I ) g ) = ( f o. g ) )
17	16	oveq1d 6665	. . . 4 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( ( f ( +g ` I ) g ) ( +g ` I ) h ) = ( ( f o. g ) ( +g ` I ) h ) )
18	8	adantr 481	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> G e. V )
19	13	3adant3r3 1276	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( f o. g ) e. ( GIsmt G ) )
20		simpr3 1069	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> h e. ( GIsmt G ) )
21	6, 7, 18, 2, 19, 20	motplusg 25437	. . . 4 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( ( f o. g ) ( +g ` I ) h ) = ( ( f o. g ) o. h ) )
22	17, 21	eqtrd 2656	. . 3 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( ( f ( +g ` I ) g ) ( +g ` I ) h ) = ( ( f o. g ) o. h ) )
23		simpr2 1068	. . . . . 6 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> g e. ( GIsmt G ) )
24	6, 7, 18, 2, 23, 20	motplusg 25437	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( g ( +g ` I ) h ) = ( g o. h ) )
25	24	oveq2d 6666	. . . 4 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( f ( +g ` I ) ( g ( +g ` I ) h ) ) = ( f ( +g ` I ) ( g o. h ) ) )
26		simpr1 1067	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> f e. ( GIsmt G ) )
27	6, 7, 18, 23, 20	motco 25435	. . . . 5 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( g o. h ) e. ( GIsmt G ) )
28	6, 7, 18, 2, 26, 27	motplusg 25437	. . . 4 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( f ( +g ` I ) ( g o. h ) ) = ( f o. ( g o. h ) ) )
29	25, 28	eqtrd 2656	. . 3 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( f ( +g ` I ) ( g ( +g ` I ) h ) ) = ( f o. ( g o. h ) ) )
30	15, 22, 29	3eqtr4a 2682	. 2 |- ( ( ph /\ ( f e. ( GIsmt G ) /\ g e. ( GIsmt G ) /\ h e. ( GIsmt G ) ) ) -> ( ( f ( +g ` I ) g ) ( +g ` I ) h ) = ( f ( +g ` I ) ( g ( +g ` I ) h ) ) )
31	6, 7, 8	idmot 25432	. 2 |- ( ph -> ( _I |` P ) e. ( GIsmt G ) )
32	8	adantr 481	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) ) -> G e. V )
33	31	adantr 481	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( _I |` P ) e. ( GIsmt G ) )
34		simpr 477	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) ) -> f e. ( GIsmt G ) )
35	6, 7, 32, 2, 33, 34	motplusg 25437	. . 3 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( ( _I |` P ) ( +g ` I ) f ) = ( ( _I |` P ) o. f ) )
36	6, 7	ismot 25430	. . . . . 6 |- ( 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 ) ) ) )
37	36	simprbda 653	. . . . 5 |- ( ( G e. V /\ f e. ( GIsmt G ) ) -> f : P -1-1-onto-> P )
38	8, 37	sylan 488	. . . 4 |- ( ( ph /\ f e. ( GIsmt G ) ) -> f : P -1-1-onto-> P )
39		f1of 6137	. . . 4 |- ( f : P -1-1-onto-> P -> f : P --> P )
40		fcoi2 6079	. . . 4 |- ( f : P --> P -> ( ( _I |` P ) o. f ) = f )
41	38, 39, 40	3syl 18	. . 3 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( ( _I |` P ) o. f ) = f )
42	35, 41	eqtrd 2656	. 2 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( ( _I |` P ) ( +g ` I ) f ) = f )
43	6, 7, 32, 34	cnvmot 25436	. 2 |- ( ( ph /\ f e. ( GIsmt G ) ) -> `' f e. ( GIsmt G ) )
44	6, 7, 32, 2, 43, 34	motplusg 25437	. . 3 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( `' f ( +g ` I ) f ) = ( `' f o. f ) )
45		f1ococnv1 6165	. . . 4 |- ( f : P -1-1-onto-> P -> ( `' f o. f ) = ( _I |` P ) )
46	38, 45	syl 17	. . 3 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( `' f o. f ) = ( _I |` P ) )
47	44, 46	eqtrd 2656	. 2 |- ( ( ph /\ f e. ( GIsmt G ) ) -> ( `' f ( +g ` I ) f ) = ( _I |` P ) )
48	4, 5, 14, 30, 31, 42, 43, 47	isgrpd 17444	1 |- ( ph -> I e. Grp )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   {cpr 4179   <.cop 4183   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   distcds 15950   Grpcgrp 17422  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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ismt 25428

This theorem is referenced by: (None)