Theorem ringressid 13169

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 12522. (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 2178	. . 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 3176	. . 3 |- ( G e. Ring -> B C_ B )
6	1, 3, 4, 5	ressbas2d 12520	. 2 |- ( G e. Ring -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2178	. . 3 |- ( G e. Ring -> ( +g ` G ) = ( +g ` G ) )
8		basfn 12512	. . . . 5 |- Base Fn _V
9		elex 2748	. . . . 5 |- ( G e. Ring -> G e. _V )
10		funfvex 5531	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5315	. . . . 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 2264	. . 3 |- ( G e. Ring -> B e. _V )
14	1, 7, 13, 4	ressplusgd 12579	. 2 |- ( G e. Ring -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		eqid 2177	. . . 4 |- ( G ↾s B ) = ( G ↾s B )
16		eqid 2177	. . . 4 |- ( .r ` G ) = ( .r ` G )
17	15, 16	ressmulrg 12595	. . 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 13115	. . 3 |- ( G e. Ring -> G e. Grp )
20	2	grpressid 12863	. . 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 13127	. 2 |- ( ( G e. Ring /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	ringass 13130	. 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 2177	. . 3 |- ( +g ` G ) = ( +g ` G )
25	2, 24, 16	ringdi 13132	. 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 13133	. 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 2177	. . 3 |- ( 1r ` G ) = ( 1r ` G )
28	2, 27	ringidcl 13134	. 2 |- ( G e. Ring -> ( 1r ` G ) e. B )
29	2, 16, 27	ringlidm 13137	. 2 |- ( ( G e. Ring /\ x e. B ) -> ( ( 1r ` G ) ( .r ` G ) x ) = x )
30	2, 16, 27	ringridm 13138	. 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 13151	1 |- ( G e. Ring -> ( G ↾s B ) e. Ring )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2737   Fn wfn 5210   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾_s cress 12455   +g cplusg 12528   .rcmulr 12529   Grpcgrp 12809   1rcur 13073   Ringcrg 13110

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-mgp 13062  df-ur 13074  df-ring 13112

This theorem is referenced by:  subrgid  13282