Theorem grpressid 13868

Description: A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13427. (Contributed by Jim Kingdon, 28-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
grpressid	⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)

Proof of Theorem grpressid

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

Step	Hyp	Ref	Expression
1		inidm 3440	. . 3 ⊢ (𝐵 ∩ 𝐵) = 𝐵
2		eqidd 2239	. . . 4 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵))
3		grpressid.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
4	3	a1i 9	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 = (Base‘𝐺))
5		id 19	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp)
6		basfn 13413	. . . . . 6 ⊢ Base Fn V
7		elex 2833	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
8		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
9	8	funfni 5483	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
10	6, 7, 9	sylancr 418	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
11	3, 10	eqeltrid 2325	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
12	2, 4, 5, 11	ressbasd 13423	. . 3 ⊢ (𝐺 ∈ Grp → (𝐵 ∩ 𝐵) = (Base‘(𝐺 ↾s 𝐵)))
13	1, 12	eqtr3id 2285	. 2 ⊢ (𝐺 ∈ Grp → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
14		eqidd 2239	. . 3 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘𝐺))
15	2, 14, 11, 7	ressplusgd 13485	. 2 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
16		eqid 2238	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	3, 16	grpcl 13815	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
18	3, 16	grpass 13816	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
19		eqid 2238	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	3, 19	grpidcl 13836	. 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	3, 16, 19	grplid 13838	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
22		eqid 2238	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	3, 22	grpinvcl 13855	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
24	3, 16, 19, 22	grplinv 13857	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
25	13, 15, 17, 18, 20, 21, 23, 24	isgrpd 13830	1 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  Basecbs 13354   ↾_s cress 13355  +_gcplusg 13433  0_gc0g 13612  Grpcgrp 13807  inv_gcminusg 13808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811

This theorem is used by:  subgid  13980  ablressid  14141  ringressid  14370