Theorem ringressid 13619

Description: A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12749. (Contributed by Jim Kingdon, 28-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
ringressid	|- ( G e. Ring -> ( G ↾s B ) e. Ring )

Proof of Theorem ringressid

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

Step	Hyp	Ref	Expression
1		eqidd 2197	. . 3 |- ( G e. Ring -> ( G ↾s B ) = ( G ↾s B ) )
2		ringressid.b	. . . 4 |- B = ( Base ` G )
3	2	a1i 9	. . 3 |- ( G e. Ring -> B = ( Base ` G ) )
4		id 19	. . 3 |- ( G e. Ring -> G e. Ring )
5		ssidd 3204	. . 3 |- ( G e. Ring -> B C_ B )
6	1, 3, 4, 5	ressbas2d 12746	. 2 |- ( G e. Ring -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2197	. . 3 |- ( G e. Ring -> ( +g ` G ) = ( +g ` G ) )
8		basfn 12736	. . . . 5 |- Base Fn _V
9		elex 2774	. . . . 5 |- ( G e. Ring -> G e. _V )
10		funfvex 5575	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5358	. . . . 5 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 414	. . . 4 |- ( G e. Ring -> ( Base ` G ) e. _V )
13	2, 12	eqeltrid 2283	. . 3 |- ( G e. Ring -> B e. _V )
14	1, 7, 13, 4	ressplusgd 12806	. 2 |- ( G e. Ring -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		eqid 2196	. . . 4 |- ( G ↾s B ) = ( G ↾s B )
16		eqid 2196	. . . 4 |- ( .r ` G ) = ( .r ` G )
17	15, 16	ressmulrg 12822	. . 3 |- ( ( B e. _V /\ G e. Ring ) -> ( .r ` G ) = ( .r ` ( G ↾s B ) ) )
18	13, 17	mpancom 422	. 2 |- ( G e. Ring -> ( .r ` G ) = ( .r ` ( G ↾s B ) ) )
19		ringgrp 13557	. . 3 |- ( G e. Ring -> G e. Grp )
20	2	grpressid 13193	. . 3 |- ( G e. Grp -> ( G ↾s B ) e. Grp )
21	19, 20	syl 14	. 2 |- ( G e. Ring -> ( G ↾s B ) e. Grp )
22	2, 16	ringcl 13569	. 2 |- ( ( G e. Ring /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	ringass 13572	. 2 |- ( ( G e. Ring /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( .r ` G ) y ) ( .r ` G ) z ) = ( x ( .r ` G ) ( y ( .r ` G ) z ) ) )
24		eqid 2196	. . 3 |- ( +g ` G ) = ( +g ` G )
25	2, 24, 16	ringdi 13574	. 2 |- ( ( G e. Ring /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x ( .r ` G ) ( y ( +g ` G ) z ) ) = ( ( x ( .r ` G ) y ) ( +g ` G ) ( x ( .r ` G ) z ) ) )
26	2, 24, 16	ringdir 13575	. 2 |- ( ( G e. Ring /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( +g ` G ) y ) ( .r ` G ) z ) = ( ( x ( .r ` G ) z ) ( +g ` G ) ( y ( .r ` G ) z ) ) )
27		eqid 2196	. . 3 |- ( 1r ` G ) = ( 1r ` G )
28	2, 27	ringidcl 13576	. 2 |- ( G e. Ring -> ( 1r ` G ) e. B )
29	2, 16, 27	ringlidm 13579	. 2 |- ( ( G e. Ring /\ x e. B ) -> ( ( 1r ` G ) ( .r ` G ) x ) = x )
30	2, 16, 27	ringridm 13580	. 2 |- ( ( G e. Ring /\ x e. B ) -> ( x ( .r ` G ) ( 1r ` G ) ) = x )
31	6, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30	isringd 13597	1 |- ( G e. Ring -> ( G ↾s B ) e. Ring )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   .rcmulr 12756   Grpcgrp 13132   1rcur 13515   Ringcrg 13552

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mgp 13477  df-ur 13516  df-ring 13554

This theorem is referenced by:  subrgid  13779