Theorem rngressid 13716

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 12903. (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 2206	. . 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 3214	. . 3 |- ( G e. Rng -> B C_ B )
6	1, 3, 4, 5	ressbas2d 12900	. 2 |- ( G e. Rng -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2206	. . 3 |- ( G e. Rng -> ( +g ` G ) = ( +g ` G ) )
8		basfn 12890	. . . . 5 |- Base Fn _V
9		elex 2783	. . . . 5 |- ( G e. Rng -> G e. _V )
10		funfvex 5593	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5376	. . . . 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 2292	. . 3 |- ( G e. Rng -> B e. _V )
14	1, 7, 13, 9	ressplusgd 12961	. 2 |- ( G e. Rng -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		eqid 2205	. . . 4 |- ( G ↾s B ) = ( G ↾s B )
16		eqid 2205	. . . 4 |- ( .r ` G ) = ( .r ` G )
17	15, 16	ressmulrg 12977	. . 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 13697	. . 3 |- ( G e. Rng -> G e. Abel )
20	2	ablressid 13671	. . 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 13706	. 2 |- ( ( G e. Rng /\ x e. B /\ y e. B ) -> ( x ( .r ` G ) y ) e. B )
23	2, 16	rngass 13701	. 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 2205	. . 3 |- ( +g ` G ) = ( +g ` G )
25	2, 24, 16	rngdi 13702	. 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 13703	. 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 13715	1 |- ( G e. Rng -> ( G ↾s B ) e. Rng )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   +g cplusg 12909   .rcmulr 12910   Abelcabl 13621  Rngcrng 13694

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-cmn 13622  df-abl 13623  df-mgp 13683  df-rng 13695

This theorem is referenced by:  subrngid  13963