Theorem ablressid 13887

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 13119. (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 2230	. . 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 3245	. . 3 |- ( G e. Abel -> B C_ B )
6	1, 3, 4, 5	ressbas2d 13116	. 2 |- ( G e. Abel -> B = ( Base ` ( G ↾s B ) ) )
7		eqidd 2230	. . 3 |- ( G e. Abel -> ( +g ` G ) = ( +g ` G ) )
8		basfn 13106	. . . . 5 |- Base Fn _V
9		elex 2811	. . . . 5 |- ( G e. Abel -> G e. _V )
10		funfvex 5646	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5423	. . . . 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 2316	. . 3 |- ( G e. Abel -> B e. _V )
14	1, 7, 13, 9	ressplusgd 13177	. 2 |- ( G e. Abel -> ( +g ` G ) = ( +g ` ( G ↾s B ) ) )
15		ablgrp 13841	. . 3 |- ( G e. Abel -> G e. Grp )
16	2	grpressid 13609	. . 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 2229	. . 3 |- ( +g ` G ) = ( +g ` G )
19	2, 18	ablcom 13855	. 2 |- ( ( G e. Abel /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
20	6, 14, 17, 19	isabld 13851	1 |- ( G e. Abel -> ( G ↾s B ) e. Abel )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   Basecbs 13047   ↾_s cress 13048   +g cplusg 13125   Grpcgrp 13548   Abelcabl 13837

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-cmn 13838  df-abl 13839

This theorem is referenced by:  rngressid  13932