Theorem ablressid 14093

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 13373. (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 2235	. . 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 3263	. . 3 |- ( G e. Abel -> B C_ B )
6	1, 3, 4, 5	ressbas2d 13370	. 2 |- ( G e. Abel -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2235	. . 3 |- ( G e. Abel -> ( +g ` G ) = ( +g ` G ) )
8		basfn 13360	. . . . 5 |- Base Fn _V
9		elex 2827	. . . . 5 |- ( G e. Abel -> G e. _V )
10		funfvex 5693	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5464	. . . . 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 2321	. . 3 |- ( G e. Abel -> B e. _V )
14	1, 7, 13, 9	ressplusgd 13431	. 2 |- ( G e. Abel -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		ablgrp 14047	. . 3 |- ( G e. Abel -> G e. Grp )
16	2	grpressid 13821	. . 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 2234	. . 3 |- ( +g ` G ) = ( +g ` G )
19	2, 18	ablcom 14061	. 2 |- ( ( G e. Abel /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
20	6, 14, 17, 19	isabld 14057	1 |- ( G e. Abel -> ( G ↾s B ) e. Abel )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   Fn wfn 5353   ` cfv 5358  (class class class)co 6059   Basecbs 13301   ↾_s cress 13302   +g cplusg 13379   Grpcgrp 13760   Abelcabl 14043

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-addass 8246  ax-i2m1 8249  ax-0lt1 8250  ax-0id 8252  ax-rnegex 8253  ax-pre-ltirr 8256  ax-pre-ltadd 8260

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-pnf 8327  df-mnf 8328  df-ltxr 8330  df-inn 9259  df-2 9317  df-ndx 13304  df-slot 13305  df-base 13307  df-sets 13308  df-iress 13309  df-plusg 13392  df-0g 13560  df-mgm 13624  df-sgrp 13670  df-mnd 13683  df-grp 13763  df-minusg 13764  df-cmn 14044  df-abl 14045

This theorem is referenced by:  rngressid  14198