Theorem rngressid 14253

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 13425. (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 13422	. 2 |- ( G e. Rng -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2239	. . 3 |- ( G e. Rng -> ( +g ` G ) = ( +g ` G ) )
8		basfn 13411	. . . . 5 |- Base Fn _V
9		elex 2833	. . . . 5 |- ( G e. Rng -> G e. _V )
10		funfvex 5712	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5483	. . . . 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 13483	. 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 13499	. . 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 14234	. . 3 |- ( G e. Rng -> G e. Abel )
20	2	ablressid 14139	. . 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 14243	. 2 |- ( ( G e. Rng /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	rngass 14238	. 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 14239	. 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 14240	. 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 14252	1 |- ( G e. Rng -> ( G ↾s B ) e. Rng )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13352   ↾_s cress 13353   +g cplusg 13431   .rcmulr 13432   Abelcabl 14088  Rngcrng 14231

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-cmn 14089  df-abl 14090  df-mgp 14218  df-rng 14232

This theorem is used by:  subrngid  14509