Theorem grpressid 12931

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 12530. (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 3345	. . 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 12520	. . . . . 6 ⊢ Base Fn V
7		elex 2749	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
8		funfvex 5533	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
9	8	funfni 5317	. . . . . 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 12527	. . 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 12587	. 2 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
16		eqid 2177	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	3, 16	grpcl 12885	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
18	3, 16	grpass 12886	. 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 12904	. 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	3, 16, 19	grplid 12906	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
22		eqid 2177	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	3, 22	grpinvcl 12921	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
24	3, 16, 19, 22	grplinv 12922	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
25	13, 15, 17, 18, 20, 21, 23, 24	isgrpd 12899	1 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2738   ∩ cin 3129   Fn wfn 5212  ‘cfv 5217  (class class class)co 5875  Basecbs 12462   ↾_s cress 12463  +_gcplusg 12536  0_gc0g 12705  Grpcgrp 12877  inv_gcminusg 12878

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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-iress 12470  df-plusg 12549  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881

This theorem is referenced by:  subgid  13035  ringressid  13238