Theorem rngressid 14115

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 13301. (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 2235	. . 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 3261	. . 3 |- ( G e. Rng -> B C_ B )
6	1, 3, 4, 5	ressbas2d 13298	. 2 |- ( G e. Rng -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2235	. . 3 |- ( G e. Rng -> ( +g ` G ) = ( +g ` G ) )
8		basfn 13288	. . . . 5 |- Base Fn _V
9		elex 2827	. . . . 5 |- ( G e. Rng -> G e. _V )
10		funfvex 5689	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5460	. . . . 5 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 414	. . . 4 |- ( G e. Rng -> ( Base ` G ) e. _V )
13	2, 12	eqeltrid 2321	. . 3 |- ( G e. Rng -> B e. _V )
14	1, 7, 13, 9	ressplusgd 13359	. 2 |- ( G e. Rng -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		eqid 2234	. . . 4 |- ( G ↾s B ) = ( G ↾s B )
16		eqid 2234	. . . 4 |- ( .r ` G ) = ( .r ` G )
17	15, 16	ressmulrg 13375	. . 3 |- ( ( B e. _V /\ G e. Rng ) -> ( .r ` G ) = ( .r ` ( G ↾s B ) ) )
18	13, 17	mpancom 422	. 2 |- ( G e. Rng -> ( .r ` G ) = ( .r ` ( G ↾s B ) ) )
19		rngabl 14096	. . 3 |- ( G e. Rng -> G e. Abel )
20	2	ablressid 14069	. . 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 14105	. 2 |- ( ( G e. Rng /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	rngass 14100	. 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 2234	. . 3 |- ( +g ` G ) = ( +g ` G )
25	2, 24, 16	rngdi 14101	. 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 14102	. 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 14114	1 |- ( G e. Rng -> ( G ↾s B ) e. Rng )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13229   ↾_s cress 13230   +g cplusg 13307   .rcmulr 13308   Abelcabl 14019  Rngcrng 14093

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-cmn 14020  df-abl 14021  df-mgp 14082  df-rng 14094

This theorem is referenced by:  subrngid  14363