Theorem ringressid 14140

Description: A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 13217. (Contributed by Jim Kingdon, 28-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
ringressid	⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)

Proof of Theorem ringressid

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2232	. . 3 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵))
2		ringressid.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	a1i 9	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘𝐺))
4		id 19	. . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Ring)
5		ssidd 3249	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 13214	. 2 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2232	. . 3 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘𝐺))
8		basfn 13204	. . . . 5 ⊢ Base Fn V
9		elex 2815	. . . . 5 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ V)
10		funfvex 5665	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5439	. . . . 5 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 414	. . . 4 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) ∈ V)
13	2, 12	eqeltrid 2318	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ∈ V)
14	1, 7, 13, 4	ressplusgd 13275	. 2 ⊢ (𝐺 ∈ Ring → (+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 13291	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐺 ∈ Ring) → (.r‘𝐺) = (.r‘(𝐺 ↾s 𝐵)))
18	13, 17	mpancom 422	. 2 ⊢ (𝐺 ∈ Ring → (.r‘𝐺) = (.r‘(𝐺 ↾s 𝐵)))
19		ringgrp 14078	. . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Grp)
20	2	grpressid 13707	. . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Grp)
22	2, 16	ringcl 14090	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	ringass 14093	. 2 ⊢ ((𝐺 ∈ Ring ∧ (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	ringdi 14095	. 2 ⊢ ((𝐺 ∈ Ring ∧ (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	ringdir 14096	. 2 ⊢ ((𝐺 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(.r‘𝐺)𝑧) = ((𝑥(.r‘𝐺)𝑧)(+g‘𝐺)(𝑦(.r‘𝐺)𝑧)))
27		eqid 2231	. . 3 ⊢ (1r‘𝐺) = (1r‘𝐺)
28	2, 27	ringidcl 14097	. 2 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ 𝐵)
29	2, 16, 27	ringlidm 14100	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((1r‘𝐺)(.r‘𝐺)𝑥) = 𝑥)
30	2, 16, 27	ringridm 14101	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝐺)(1r‘𝐺)) = 𝑥)
31	6, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30	isringd 14118	1 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  Basecbs 13145   ↾_s cress 13146  +_gcplusg 13223  ._rcmulr 13224  Grpcgrp 13646  1_rcur 14036  Ringcrg 14073

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-mgp 13998  df-ur 14037  df-ring 14075

This theorem is referenced by:  subrgid  14301