Theorem rngressid 13831

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 13018. (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 2208	. . 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 3222	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 13015	. 2 ⊢ (𝐺 ∈ Rng → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2208	. . 3 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘𝐺))
8		basfn 13005	. . . . 5 ⊢ Base Fn V
9		elex 2788	. . . . 5 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ V)
10		funfvex 5616	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5395	. . . . 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 2294	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ∈ V)
14	1, 7, 13, 9	ressplusgd 13076	. 2 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
15		eqid 2207	. . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)
16		eqid 2207	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
17	15, 16	ressmulrg 13092	. . 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 13812	. . 3 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ Abel)
20	2	ablressid 13786	. . 3 ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Abel)
22	2, 16	rngcl 13821	. 2 ⊢ ((𝐺 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	rngass 13816	. 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 2207	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	2, 24, 16	rngdi 13817	. 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 13818	. 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 13830	1 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178  Vcvv 2776   Fn wfn 5285  ‘cfv 5290  (class class class)co 5967  Basecbs 12947   ↾_s cress 12948  +_gcplusg 13024  ._rcmulr 13025  Abelcabl 13736  Rngcrng 13809

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-cmn 13737  df-abl 13738  df-mgp 13798  df-rng 13810

This theorem is referenced by:  subrngid  14078