Theorem rngressid 13957

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 13144. (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 2230	. . 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 3246	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 13141	. 2 ⊢ (𝐺 ∈ Rng → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2230	. . 3 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘𝐺))
8		basfn 13131	. . . . 5 ⊢ Base Fn V
9		elex 2812	. . . . 5 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ V)
10		funfvex 5652	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5429	. . . . 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 2316	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ∈ V)
14	1, 7, 13, 9	ressplusgd 13202	. 2 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
15		eqid 2229	. . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)
16		eqid 2229	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
17	15, 16	ressmulrg 13218	. . 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 13938	. . 3 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ Abel)
20	2	ablressid 13912	. . 3 ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Abel)
22	2, 16	rngcl 13947	. 2 ⊢ ((𝐺 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	rngass 13942	. 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 2229	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	2, 24, 16	rngdi 13943	. 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 13944	. 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 13956	1 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2800   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013  Basecbs 13072   ↾_s cress 13073  +_gcplusg 13150  ._rcmulr 13151  Abelcabl 13862  Rngcrng 13935

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-cmn 13863  df-abl 13864  df-mgp 13924  df-rng 13936

This theorem is referenced by:  subrngid  14205