Theorem ringressid 14306

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 13368. (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 2235	. . 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 3263	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 13365	. 2 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2235	. . 3 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘𝐺))
8		basfn 13355	. . . . 5 ⊢ Base Fn V
9		elex 2827	. . . . 5 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ V)
10		funfvex 5692	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5463	. . . . 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 2321	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ∈ V)
14	1, 7, 13, 4	ressplusgd 13426	. 2 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
15		eqid 2234	. . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)
16		eqid 2234	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
17	15, 16	ressmulrg 13442	. . 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 14244	. . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Grp)
20	2	grpressid 13816	. . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Grp)
22	2, 16	ringcl 14256	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	ringass 14259	. 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 2234	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	2, 24, 16	ringdi 14261	. 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 14262	. 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 2234	. . 3 ⊢ (1r‘𝐺) = (1r‘𝐺)
28	2, 27	ringidcl 14263	. 2 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ 𝐵)
29	2, 16, 27	ringlidm 14266	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((1r‘𝐺)(.r‘𝐺)𝑥) = 𝑥)
30	2, 16, 27	ringridm 14267	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝐺)(1r‘𝐺)) = 𝑥)
31	6, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30	isringd 14284	1 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815   Fn wfn 5352  ‘cfv 5357  (class class class)co 6058  Basecbs 13296   ↾_s cress 13297  +_gcplusg 13374  ._rcmulr 13375  Grpcgrp 13755  1_rcur 14202  Ringcrg 14239

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-mgp 14160  df-ur 14203  df-ring 14241

This theorem is referenced by:  subrgid  14469