Theorem grpressid 13763

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 13273. (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 3429	. . 3 ⊢ (𝐵 ∩ 𝐵) = 𝐵
2		eqidd 2233	. . . 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 13260	. . . . . 6 ⊢ Base Fn V
7		elex 2824	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
8		funfvex 5686	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
9	8	funfni 5457	. . . . . 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 2319	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
12	2, 4, 5, 11	ressbasd 13269	. . 3 ⊢ (𝐺 ∈ Grp → (𝐵 ∩ 𝐵) = (Base‘(𝐺 ↾s 𝐵)))
13	1, 12	eqtr3id 2279	. 2 ⊢ (𝐺 ∈ Grp → 𝐵 = (Base‘(𝐺 ↾s 𝐵)))
14		eqidd 2233	. . 3 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘𝐺))
15	2, 14, 11, 7	ressplusgd 13331	. 2 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵)))
16		eqid 2232	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	3, 16	grpcl 13710	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
18	3, 16	grpass 13711	. 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 2232	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	3, 19	grpidcl 13731	. 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	3, 16, 19	grplid 13733	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
22		eqid 2232	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	3, 22	grpinvcl 13750	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
24	3, 16, 19, 22	grplinv 13752	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
25	13, 15, 17, 18, 20, 21, 23, 24	isgrpd 13725	1 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ∩ cin 3209   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049  Basecbs 13201   ↾_s cress 13202  +_gcplusg 13279  0_gc0g 13458  Grpcgrp 13702  inv_gcminusg 13703

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-minusg 13706

This theorem is referenced by:  subgid  13881  ablressid  14041  ringressid  14196