Theorem grpressid 13843

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 13402. (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 13389	. . . . . 6 |- Base Fn _V
7		elex 2833	. . . . . 6 |- ( G e. Grp -> G e. _V )
8		funfvex 5707	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
9	8	funfni 5478	. . . . . 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 13398	. . 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 13460	. 2 |- ( G e. Grp -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
16		eqid 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
17	3, 16	grpcl 13790	. 2 |- ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
18	3, 16	grpass 13791	. 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 13811	. 2 |- ( G e. Grp -> ( 0g ` G ) e. B )
21	3, 16, 19	grplid 13813	. 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 13830	. 2 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
24	3, 16, 19, 22	grplinv 13832	. 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 13805	1 |- ( G e. Grp -> ( G ↾s B ) e. Grp )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782   invgcminusg 13783

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786

This theorem is referenced by:  subgid  13955  ablressid  14116  ringressid  14341