Theorem grpressid 12936

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 12532. (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 3346	. . 3 ⊢ (𝐵 ∩ 𝐵) = 𝐵
2		eqidd 2178	. . . 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 12522	. . . . . 6 ⊢ Base Fn V
7		elex 2750	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
8		funfvex 5534	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
9	8	funfni 5318	. . . . . 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 2264	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
12	2, 4, 5, 11	ressbasd 12529	. . 3 ⊢ (𝐺 ∈ Grp → (𝐵 ∩ 𝐵) = (Base‘(𝐺 ↾s 𝐵)))
13	1, 12	eqtr3id 2224	. 2 ⊢ (𝐺 ∈ Grp → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
14		eqidd 2178	. . 3 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘𝐺))
15	2, 14, 11, 7	ressplusgd 12589	. 2 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
16		eqid 2177	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	3, 16	grpcl 12890	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
18	3, 16	grpass 12891	. 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 2177	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	3, 19	grpidcl 12909	. 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	3, 16, 19	grplid 12911	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
22		eqid 2177	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	3, 22	grpinvcl 12926	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
24	3, 16, 19, 22	grplinv 12927	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
25	13, 15, 17, 18, 20, 21, 23, 24	isgrpd 12904	1 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∩ cin 3130   Fn wfn 5213  ‘cfv 5218  (class class class)co 5877  Basecbs 12464   ↾_s cress 12465  +_gcplusg 12538  0_gc0g 12710  Grpcgrp 12882  inv_gcminusg 12883

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 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886

This theorem is referenced by:  subgid  13040  ringressid  13243