Theorem grpressid 13637

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 13147. (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 3414	. . 3 ⊢ (𝐵 ∩ 𝐵) = 𝐵
2		eqidd 2230	. . . 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 13134	. . . . . 6 ⊢ Base Fn V
7		elex 2812	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
8		funfvex 5652	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
9	8	funfni 5429	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
10	6, 7, 9	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
11	3, 10	eqeltrid 2316	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
12	2, 4, 5, 11	ressbasd 13143	. . 3 ⊢ (𝐺 ∈ Grp → (𝐵 ∩ 𝐵) = (Base‘(𝐺 ↾s 𝐵)))
13	1, 12	eqtr3id 2276	. 2 ⊢ (𝐺 ∈ Grp → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
14		eqidd 2230	. . 3 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘𝐺))
15	2, 14, 11, 7	ressplusgd 13205	. 2 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
16		eqid 2229	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	3, 16	grpcl 13584	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
18	3, 16	grpass 13585	. 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 2229	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	3, 19	grpidcl 13605	. 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	3, 16, 19	grplid 13607	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
22		eqid 2229	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	3, 22	grpinvcl 13624	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
24	3, 16, 19, 22	grplinv 13626	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
25	13, 15, 17, 18, 20, 21, 23, 24	isgrpd 13599	1 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∩ cin 3197   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013  Basecbs 13075   ↾_s cress 13076  +_gcplusg 13153  0_gc0g 13332  Grpcgrp 13576  inv_gcminusg 13577

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-grp 13579  df-minusg 13580

This theorem is referenced by:  subgid  13755  ablressid  13915  ringressid  14069