Theorem grpressid 13335

Description: A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12845. (Contributed by Jim Kingdon, 28-Feb-2025.)

Hypothesis

Ref	Expression
grpressid.b	|- B = ( Base ` G )

Assertion

Ref	Expression
grpressid	|- ( G e. Grp -> ( G ↾s B ) e. Grp )

Proof of Theorem grpressid

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inidm 3381	. . 3 |- ( B i^i B ) = B
2		eqidd 2205	. . . 4 |- ( G e. Grp -> ( G ↾s B ) = ( G ↾s B ) )
3		grpressid.b	. . . . 5 |- B = ( Base ` G )
4	3	a1i 9	. . . 4 |- ( G e. Grp -> B = ( Base ` G ) )
5		id 19	. . . 4 |- ( G e. Grp -> G e. Grp )
6		basfn 12832	. . . . . 6 |- Base Fn _V
7		elex 2782	. . . . . 6 |- ( G e. Grp -> G e. _V )
8		funfvex 5592	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
9	8	funfni 5375	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
10	6, 7, 9	sylancr 414	. . . . 5 |- ( G e. Grp -> ( Base ` G ) e. _V )
11	3, 10	eqeltrid 2291	. . . 4 |- ( G e. Grp -> B e. _V )
12	2, 4, 5, 11	ressbasd 12841	. . 3 |- ( G e. Grp -> ( B i^i B ) = ( Base ` ( G ↾s B ) ) )
13	1, 12	eqtr3id 2251	. 2 |- ( G e. Grp -> B = ( Base ` ( G ↾s B ) ) )
14		eqidd 2205	. . 3 |- ( G e. Grp -> ( +g ` G ) = ( +g ` G ) )
15	2, 14, 11, 7	ressplusgd 12903	. 2 |- ( G e. Grp -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
16		eqid 2204	. . 3 |- ( +g ` G ) = ( +g ` G )
17	3, 16	grpcl 13282	. 2 |- ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
18	3, 16	grpass 13283	. 2 |- ( ( G e. Grp /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) )
19		eqid 2204	. . 3 |- ( 0g ` G ) = ( 0g ` G )
20	3, 19	grpidcl 13303	. 2 |- ( G e. Grp -> ( 0g ` G ) e. B )
21	3, 16, 19	grplid 13305	. 2 |- ( ( G e. Grp /\ x e. B ) -> ( ( 0g ` G ) ( +g ` G ) x ) = x )
22		eqid 2204	. . 3 |- ( invg ` G ) = ( invg ` G )
23	3, 22	grpinvcl 13322	. 2 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
24	3, 16, 19, 22	grplinv 13324	. 2 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( invg ` G ) ` x ) ( +g ` G ) x ) = ( 0g ` G ) )
25	13, 15, 17, 18, 20, 21, 23, 24	isgrpd 13297	1 |- ( G e. Grp -> ( G ↾s B ) e. Grp )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   _Vcvv 2771   i^i cin 3164   Fn wfn 5265   ` cfv 5270  (class class class)co 5943   Basecbs 12774   ↾_s cress 12775   +g cplusg 12851   0gc0g 13030   Grpcgrp 13274   invgcminusg 13275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-iress 12782  df-plusg 12864  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278

This theorem is referenced by:  subgid  13453  ablressid  13613  ringressid  13767