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Theorem motgrp 23906
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.
StepHypRef Expression
1 ovex 6309 . . 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 ) ) >. }
32grpbase 14718 . . 3  |-  ( ( GIsmt G )  e. 
_V  ->  ( GIsmt G
)  =  ( Base `  I ) )
41, 3mp1i 12 . 2  |-  ( ph  ->  ( GIsmt G )  =  ( Base `  I
) )
5 eqidd 2444 . 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 )
983ad2ant1 1018 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G ) )  ->  G  e.  V )
10 simp2 998 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G ) )  -> 
f  e.  ( GIsmt G ) )
11 simp3 999 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G ) )  -> 
g  e.  ( GIsmt G ) )
126, 7, 9, 2, 10, 11motplusg 23905 . . 3  |-  ( (
ph  /\  f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G ) )  -> 
( f ( +g  `  I ) g )  =  ( f  o.  g ) )
136, 7, 9, 10, 11motco 23903 . . 3  |-  ( (
ph  /\  f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G ) )  -> 
( f  o.  g
)  e.  ( GIsmt G ) )
1412, 13eqeltrd 2531 . 2  |-  ( (
ph  /\  f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G ) )  -> 
( f ( +g  `  I ) g )  e.  ( GIsmt G
) )
15 coass 5516 . . 3  |-  ( ( f  o.  g )  o.  h )  =  ( f  o.  (
g  o.  h ) )
16123adant3r3 1208 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  ( f ( +g  `  I ) g )  =  ( f  o.  g ) )
1716oveq1d 6296 . . . 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 ) )
188adantr 465 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  G  e.  V
)
19133adant3r3 1208 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  ( f  o.  g )  e.  ( GIsmt G ) )
20 simpr3 1005 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  h  e.  ( GIsmt G ) )
216, 7, 18, 2, 19, 20motplusg 23905 . . . 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 ) )
2217, 21eqtrd 2484 . . 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 1004 . . . . . 6  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  g  e.  ( GIsmt G ) )
246, 7, 18, 2, 23, 20motplusg 23905 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  ( g ( +g  `  I ) h )  =  ( g  o.  h ) )
2524oveq2d 6297 . . . 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 1003 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  f  e.  ( GIsmt G ) )
276, 7, 18, 23, 20motco 23903 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( GIsmt G )  /\  g  e.  ( GIsmt G )  /\  h  e.  ( GIsmt G ) ) )  ->  ( g  o.  h )  e.  ( GIsmt G ) )
286, 7, 18, 2, 26, 27motplusg 23905 . . . 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 ) ) )
2925, 28eqtrd 2484 . . 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 ) ) )
3015, 22, 293eqtr4a 2510 . 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 ) ) )
316, 7, 8idmot 23900 . 2  |-  ( ph  ->  (  _I  |`  P )  e.  ( GIsmt G
) )
328adantr 465 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  G  e.  V
)
3331adantr 465 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  (  _I  |`  P )  e.  ( GIsmt G
) )
34 simpr 461 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  f  e.  ( GIsmt G ) )
356, 7, 32, 2, 33, 34motplusg 23905 . . 3  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  ( (  _I  |`  P ) ( +g  `  I ) f )  =  ( (  _I  |`  P )  o.  f
) )
366, 7ismot 23898 . . . . . 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 ) ) ) )
3736simprbda 623 . . . . 5  |-  ( ( G  e.  V  /\  f  e.  ( GIsmt G ) )  -> 
f : P -1-1-onto-> P )
388, 37sylan 471 . . . 4  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  f : P -1-1-onto-> P
)
39 f1of 5806 . . . 4  |-  ( f : P -1-1-onto-> P  ->  f : P
--> P )
40 fcoi2 5750 . . . 4  |-  ( f : P --> P  -> 
( (  _I  |`  P )  o.  f )  =  f )
4138, 39, 403syl 20 . . 3  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  ( (  _I  |`  P )  o.  f
)  =  f )
4235, 41eqtrd 2484 . 2  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  ( (  _I  |`  P ) ( +g  `  I ) f )  =  f )
436, 7, 32, 34cnvmot 23904 . 2  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  `' f  e.  ( GIsmt G ) )
446, 7, 32, 2, 43, 34motplusg 23905 . . 3  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  ( `' f ( +g  `  I
) f )  =  ( `' f  o.  f ) )
45 f1ococnv1 5834 . . . 4  |-  ( f : P -1-1-onto-> P  ->  ( `' f  o.  f )  =  (  _I  |`  P ) )
4638, 45syl 16 . . 3  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  ( `' f  o.  f )  =  (  _I  |`  P ) )
4744, 46eqtrd 2484 . 2  |-  ( (
ph  /\  f  e.  ( GIsmt G ) )  ->  ( `' f ( +g  `  I
) f )  =  (  _I  |`  P ) )
484, 5, 14, 30, 31, 42, 43, 47isgrpd 16053 1  |-  ( ph  ->  I  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095   {cpr 4016   <.cop 4020    _I cid 4780   `'ccnv 4988    |` cres 4991    o. ccom 4993   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   ndxcnx 14610   Basecbs 14613   +g cplusg 14678   distcds 14687   Grpcgrp 16031  Ismtcismt 23895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11092  df-fz 11683  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-plusg 14691  df-0g 14820  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035  df-ismt 23896
This theorem is referenced by: (None)
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