Theorem rngressid 13966

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 13153. (Contributed by Jim Kingdon, 5-May-2025.)

Hypothesis

Ref	Expression
rngressid.b	⊢ 𝐵 = (Base‘𝐺)

Assertion

Ref	Expression
rngressid	⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ 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 2232	. . 3 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵))
2		rngressid.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	a1i 9	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 = (Base‘𝐺))
4		id 19	. . 3 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ Rng)
5		ssidd 3248	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 13150	. 2 ⊢ (𝐺 ∈ Rng → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2232	. . 3 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘𝐺))
8		basfn 13140	. . . . 5 ⊢ Base Fn V
9		elex 2814	. . . . 5 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ V)
10		funfvex 5656	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5432	. . . . 5 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 414	. . . 4 ⊢ (𝐺 ∈ Rng → (Base‘𝐺) ∈ V)
13	2, 12	eqeltrid 2318	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ∈ V)
14	1, 7, 13, 9	ressplusgd 13211	. 2 ⊢ (𝐺 ∈ Rng → (+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 13227	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐺 ∈ Rng) → (.r‘𝐺) = (.r‘(𝐺 ↾s 𝐵)))
18	13, 17	mpancom 422	. 2 ⊢ (𝐺 ∈ Rng → (.r‘𝐺) = (.r‘(𝐺 ↾s 𝐵)))
19		rngabl 13947	. . 3 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ Abel)
20	2	ablressid 13921	. . 3 ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Abel)
22	2, 16	rngcl 13956	. 2 ⊢ ((𝐺 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	rngass 13951	. 2 ⊢ ((𝐺 ∈ Rng ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 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	rngdi 13952	. 2 ⊢ ((𝐺 ∈ Rng ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥(.r‘𝐺)(𝑦(+g‘𝐺)𝑧)) = ((𝑥(.r‘𝐺)𝑦)(+g‘𝐺)(𝑥(.r‘𝐺)𝑧)))
26	2, 24, 16	rngdir 13953	. 2 ⊢ ((𝐺 ∈ Rng ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 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 13965	1 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017  Basecbs 13081   ↾_s cress 13082  +_gcplusg 13159  ._rcmulr 13160  Abelcabl 13871  Rngcrng 13944

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-rng 13945

This theorem is referenced by:  subrngid  14214