Theorem ringressid 14207

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 13284. (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 2233	. . 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 3259	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 13281	. 2 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2233	. . 3 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘𝐺))
8		basfn 13271	. . . . 5 ⊢ Base Fn V
9		elex 2825	. . . . 5 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ V)
10		funfvex 5687	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5458	. . . . 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 2319	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ∈ V)
14	1, 7, 13, 4	ressplusgd 13342	. 2 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
15		eqid 2232	. . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)
16		eqid 2232	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
17	15, 16	ressmulrg 13358	. . 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 14145	. . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Grp)
20	2	grpressid 13774	. . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Grp)
22	2, 16	ringcl 14157	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	ringass 14160	. 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 2232	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	2, 24, 16	ringdi 14162	. 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 14163	. 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 2232	. . 3 ⊢ (1r‘𝐺) = (1r‘𝐺)
28	2, 27	ringidcl 14164	. 2 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ 𝐵)
29	2, 16, 27	ringlidm 14167	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((1r‘𝐺)(.r‘𝐺)𝑥) = 𝑥)
30	2, 16, 27	ringridm 14168	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝐺)(1r‘𝐺)) = 𝑥)
31	6, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30	isringd 14185	1 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  Basecbs 13212   ↾_s cress 13213  +_gcplusg 13290  ._rcmulr 13291  Grpcgrp 13713  1_rcur 14103  Ringcrg 14140

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-mgp 14065  df-ur 14104  df-ring 14142

This theorem is referenced by:  subrgid  14368