Theorem rngressid 13658

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 12845. (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 2205	. . 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 3213	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 12842	. 2 ⊢ (𝐺 ∈ Rng → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2205	. . 3 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘𝐺))
8		basfn 12832	. . . . 5 ⊢ Base Fn V
9		elex 2782	. . . . 5 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ V)
10		funfvex 5592	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5375	. . . . 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 2291	. . 3 ⊢ (𝐺 ∈ Rng → 𝐵 ∈ V)
14	1, 7, 13, 9	ressplusgd 12903	. 2 ⊢ (𝐺 ∈ Rng → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
15		eqid 2204	. . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)
16		eqid 2204	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
17	15, 16	ressmulrg 12919	. . 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 13639	. . 3 ⊢ (𝐺 ∈ Rng → 𝐺 ∈ Abel)
20	2	ablressid 13613	. . 3 ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Abel)
22	2, 16	rngcl 13648	. 2 ⊢ ((𝐺 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	rngass 13643	. 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 2204	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	2, 24, 16	rngdi 13644	. 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 13645	. 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 13657	1 ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)

Syntax hints:   → wi 4   = wceq 1372   ∈ wcel 2175  Vcvv 2771   Fn wfn 5265  ‘cfv 5270  (class class class)co 5943  Basecbs 12774   ↾_s cress 12775  +_gcplusg 12851  ._rcmulr 12852  Abelcabl 13563  Rngcrng 13636

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-iress 12782  df-plusg 12864  df-mulr 12865  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278  df-cmn 13564  df-abl 13565  df-mgp 13625  df-rng 13637

This theorem is referenced by:  subrngid  13905