Theorem grpressid 13919

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 13478. (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 3440	. . 3 |- ( B i^i B ) = B
2		eqidd 2239	. . . 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 13463	. . . . . 6 |- Base Fn _V
7		elex 2833	. . . . . 6 |- ( G e. Grp -> G e. _V )
8		funfvex 5712	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
9	8	funfni 5483	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
10	6, 7, 9	sylancr 418	. . . . 5 |- ( G e. Grp -> ( Base ` G ) e. _V )
11	3, 10	eqeltrid 2325	. . . 4 |- ( G e. Grp -> B e. _V )
12	2, 4, 5, 11	ressbasd 13474	. . 3 |- ( G e. Grp -> ( B i^i B ) = ( Base ` ( G ↾s B ) ) )
13	1, 12	eqtr3id 2285	. 2 |- ( G e. Grp -> B = ( Base ` ( G ↾s B ) ) )
14		eqidd 2239	. . 3 |- ( G e. Grp -> ( +g ` G ) = ( +g ` G ) )
15	2, 14, 11, 7	ressplusgd 13536	. 2 |- ( G e. Grp -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
16		eqid 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
17	3, 16	grpcl 13866	. 2 |- ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
18	3, 16	grpass 13867	. 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 2238	. . 3 |- ( 0g ` G ) = ( 0g ` G )
20	3, 19	grpidcl 13887	. 2 |- ( G e. Grp -> ( 0g ` G ) e. B )
21	3, 16, 19	grplid 13889	. 2 |- ( ( G e. Grp /\ x e. B ) -> ( ( 0g ` G ) ( +g ` G ) x ) = x )
22		eqid 2238	. . 3 |- ( invg ` G ) = ( invg ` G )
23	3, 22	grpinvcl 13906	. 2 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
24	3, 16, 19, 22	grplinv 13908	. 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 13881	1 |- ( G e. Grp -> ( G ↾s B ) e. Grp )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13404   ↾_s cress 13405   +g cplusg 13484   0gc0g 13663   Grpcgrp 13858   invgcminusg 13859

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-iress 13412  df-plusg 13497  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-grp 13861  df-minusg 13862

This theorem is used by:  subgid  14031  ablressid  14223  ringressid  14452