Theorem ringressid 13191

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 12524. (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 2178	. . 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 3176	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ⊆ 𝐵)
6	1, 3, 4, 5	ressbas2d 12522	. 2 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
7		eqidd 2178	. . 3 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘𝐺))
8		basfn 12514	. . . . 5 ⊢ Base Fn V
9		elex 2748	. . . . 5 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ V)
10		funfvex 5532	. . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5316	. . . . 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 2264	. . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ∈ V)
14	1, 7, 13, 4	ressplusgd 12581	. 2 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
15		eqid 2177	. . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)
16		eqid 2177	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
17	15, 16	ressmulrg 12597	. . 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 13137	. . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Grp)
20	2	grpressid 12885	. . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
21	19, 20	syl 14	. 2 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Grp)
22	2, 16	ringcl 13149	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵)
23	2, 16	ringass 13152	. 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 2177	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	2, 24, 16	ringdi 13154	. 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 13155	. 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 2177	. . 3 ⊢ (1r‘𝐺) = (1r‘𝐺)
28	2, 27	ringidcl 13156	. 2 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ 𝐵)
29	2, 16, 27	ringlidm 13159	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((1r‘𝐺)(.r‘𝐺)𝑥) = 𝑥)
30	2, 16, 27	ringridm 13160	. 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝐺)(1r‘𝐺)) = 𝑥)
31	6, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30	isringd 13173	1 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2737   Fn wfn 5211  ‘cfv 5216  (class class class)co 5874  Basecbs 12456   ↾_s cress 12457  +_gcplusg 12530  ._rcmulr 12531  Grpcgrp 12831  1_rcur 13095  Ringcrg 13132

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-mgp 13084  df-ur 13096  df-ring 13134

This theorem is referenced by:  subrgid  13304