Theorem ablressid 13541

Description: A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12774. (Contributed by Jim Kingdon, 5-May-2025.)

Hypothesis

Ref	Expression
ablressid.b	|- B = ( Base ` G )

Assertion

Ref	Expression
ablressid	|- ( G e. Abel -> ( G ↾s B ) e. Abel )

Proof of Theorem ablressid

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

Step	Hyp	Ref	Expression
1		eqidd 2197	. . 3 |- ( G e. Abel -> ( G ↾s B ) = ( G ↾s B ) )
2		ablressid.b	. . . 4 |- B = ( Base ` G )
3	2	a1i 9	. . 3 |- ( G e. Abel -> B = ( Base ` G ) )
4		id 19	. . 3 |- ( G e. Abel -> G e. Abel )
5		ssidd 3205	. . 3 |- ( G e. Abel -> B C_ B )
6	1, 3, 4, 5	ressbas2d 12771	. 2 |- ( G e. Abel -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2197	. . 3 |- ( G e. Abel -> ( +g ` G ) = ( +g ` G ) )
8		basfn 12761	. . . . 5 |- Base Fn _V
9		elex 2774	. . . . 5 |- ( G e. Abel -> G e. _V )
10		funfvex 5578	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5361	. . . . 5 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 414	. . . 4 |- ( G e. Abel -> ( Base ` G ) e. _V )
13	2, 12	eqeltrid 2283	. . 3 |- ( G e. Abel -> B e. _V )
14	1, 7, 13, 9	ressplusgd 12831	. 2 |- ( G e. Abel -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		ablgrp 13495	. . 3 |- ( G e. Abel -> G e. Grp )
16	2	grpressid 13263	. . 3 |- ( G e. Grp -> ( G ↾s B ) e. Grp )
17	15, 16	syl 14	. 2 |- ( G e. Abel -> ( G ↾s B ) e. Grp )
18		eqid 2196	. . 3 |- ( +g ` G ) = ( +g ` G )
19	2, 18	ablcom 13509	. 2 |- ( ( G e. Abel /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
20	6, 14, 17, 19	isabld 13505	1 |- ( G e. Abel -> ( G ↾s B ) e. Abel )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   Grpcgrp 13202   Abelcabl 13491

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-cmn 13492  df-abl 13493

This theorem is referenced by:  rngressid  13586