Theorem rngressid 14231

Description: A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 13405. (Contributed by Jim Kingdon, 5-May-2025.)

Hypothesis

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

Assertion

Ref	Expression
rngressid	|- ( G e. Rng -> ( G ↾s B ) e. Rng )

Proof of Theorem rngressid

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

Step	Hyp	Ref	Expression
1		eqidd 2239	. . 3 |- ( G e. Rng -> ( G ↾s B ) = ( G ↾s B ) )
2		rngressid.b	. . . 4 |- B = ( Base ` G )
3	2	a1i 9	. . 3 |- ( G e. Rng -> B = ( Base ` G ) )
4		id 19	. . 3 |- ( G e. Rng -> G e. Rng )
5		ssidd 3269	. . 3 |- ( G e. Rng -> B C_ B )
6	1, 3, 4, 5	ressbas2d 13402	. 2 |- ( G e. Rng -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2239	. . 3 |- ( G e. Rng -> ( +g ` G ) = ( +g ` G ) )
8		basfn 13392	. . . . 5 |- Base Fn _V
9		elex 2833	. . . . 5 |- ( G e. Rng -> G e. _V )
10		funfvex 5710	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5481	. . . . 5 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 418	. . . 4 |- ( G e. Rng -> ( Base ` G ) e. _V )
13	2, 12	eqeltrid 2325	. . 3 |- ( G e. Rng -> B e. _V )
14	1, 7, 13, 9	ressplusgd 13463	. 2 |- ( G e. Rng -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		eqid 2238	. . . 4 |- ( G ↾s B ) = ( G ↾s B )
16		eqid 2238	. . . 4 |- ( .r ` G ) = ( .r ` G )
17	15, 16	ressmulrg 13479	. . 3 |- ( ( B e. _V /\ G e. Rng ) -> ( .r ` G ) = ( .r ` ( G ↾s B ) ) )
18	13, 17	mpancom 426	. 2 |- ( G e. Rng -> ( .r ` G ) = ( .r ` ( G ↾s B ) ) )
19		rngabl 14212	. . 3 |- ( G e. Rng -> G e. Abel )
20	2	ablressid 14119	. . 3 |- ( G e. Abel -> ( G ↾s B ) e. Abel )
21	19, 20	syl 14	. 2 |- ( G e. Rng -> ( G ↾s B ) e. Abel )
22	2, 16	rngcl 14221	. 2 |- ( ( G e. Rng /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	rngass 14216	. 2 |- ( ( G e. Rng /\ ( 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 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
25	2, 24, 16	rngdi 14217	. 2 |- ( ( G e. Rng /\ ( 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	rngdir 14218	. 2 |- ( ( G e. Rng /\ ( 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	6, 14, 18, 21, 22, 23, 25, 26	isrngd 14230	1 |- ( G e. Rng -> ( G ↾s B ) e. Rng )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   Fn wfn 5370   ` cfv 5375  (class class class)co 6078   Basecbs 13333   ↾_s cress 13334   +g cplusg 13411   .rcmulr 13412   Abelcabl 14068  Rngcrng 14209

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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-i2m1 8277  ax-0lt1 8278  ax-0id 8280  ax-rnegex 8281  ax-pre-ltirr 8284  ax-pre-lttrn 8286  ax-pre-ltadd 8288

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-pnf 8355  df-mnf 8356  df-ltxr 8358  df-inn 9287  df-2 9345  df-3 9346  df-ndx 13336  df-slot 13337  df-base 13339  df-sets 13340  df-iress 13341  df-plusg 13424  df-mulr 13425  df-0g 13592  df-mgm 13656  df-sgrp 13697  df-mnd 13710  df-grp 13788  df-minusg 13789  df-cmn 14069  df-abl 14070  df-mgp 14198  df-rng 14210

This theorem is referenced by:  subrngid  14485