Theorem ringressid 14157

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 13234. (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 2232	. . 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 3249	. . 3 |- ( G e. Ring -> B C_ B )
6	1, 3, 4, 5	ressbas2d 13231	. 2 |- ( G e. Ring -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2232	. . 3 |- ( G e. Ring -> ( +g ` G ) = ( +g ` G ) )
8		basfn 13221	. . . . 5 |- Base Fn _V
9		elex 2815	. . . . 5 |- ( G e. Ring -> G e. _V )
10		funfvex 5665	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5439	. . . . 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 2318	. . 3 |- ( G e. Ring -> B e. _V )
14	1, 7, 13, 4	ressplusgd 13292	. 2 |- ( G e. Ring -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		eqid 2231	. . . 4 |- ( G ↾s B ) = ( G ↾s B )
16		eqid 2231	. . . 4 |- ( .r ` G ) = ( .r ` G )
17	15, 16	ressmulrg 13308	. . 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 14095	. . 3 |- ( G e. Ring -> G e. Grp )
20	2	grpressid 13724	. . 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 14107	. 2 |- ( ( G e. Ring /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	ringass 14110	. 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 2231	. . 3 |- ( +g ` G ) = ( +g ` G )
25	2, 24, 16	ringdi 14112	. 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 14113	. 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 2231	. . 3 |- ( 1r ` G ) = ( 1r ` G )
28	2, 27	ringidcl 14114	. 2 |- ( G e. Ring -> ( 1r ` G ) e. B )
29	2, 16, 27	ringlidm 14117	. 2 |- ( ( G e. Ring /\ x e. B ) -> ( ( 1r ` G ) ( .r ` G ) x ) = x )
30	2, 16, 27	ringridm 14118	. 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 14135	1 |- ( G e. Ring -> ( G ↾s B ) e. Ring )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   Basecbs 13162   ↾_s cress 13163   +g cplusg 13240   .rcmulr 13241   Grpcgrp 13663   1rcur 14053   Ringcrg 14090

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-mgp 14015  df-ur 14054  df-ring 14092

This theorem is referenced by:  subrgid  14318