Theorem grpss 17440

Description: Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows us to prove that a constructed potential ring R is a group before we know that it is also a ring. (Theorem ringgrp 18552, on the other hand, requires that we know in advance that R is a ring.) (Contributed by NM, 11-Oct-2013.)

Hypotheses

Ref	Expression
grpss.g	|- G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }
grpss.r	|- R e. _V
grpss.s	|- G C_ R
grpss.f	|- Fun R

Assertion

Ref	Expression
grpss	|- ( G e. Grp <-> R e. Grp )

Proof of Theorem grpss

Step	Hyp	Ref	Expression
1		grpss.r	. . . 4 |- R e. _V
2		grpss.f	. . . 4 |- Fun R
3		grpss.s	. . . 4 |- G C_ R
4		baseid 15919	. . . 4 |- Base = Slot ( Base ` ndx )
5		opex 4932	. . . . . 6 |- <. ( Base ` ndx ) , B >. e. _V
6	5	prid1 4297	. . . . 5 |- <. ( Base ` ndx ) , B >. e. { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }
7		grpss.g	. . . . 5 |- G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }
8	6, 7	eleqtrri 2700	. . . 4 |- <. ( Base ` ndx ) , B >. e. G
9	1, 2, 3, 4, 8	strss 15910	. . 3 |- ( Base ` R ) = ( Base ` G )
10		plusgid 15977	. . . 4 |- +g = Slot ( +g ` ndx )
11		opex 4932	. . . . . 6 |- <. ( +g ` ndx ) , .+ >. e. _V
12	11	prid2 4298	. . . . 5 |- <. ( +g ` ndx ) , .+ >. e. { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }
13	12, 7	eleqtrri 2700	. . . 4 |- <. ( +g ` ndx ) , .+ >. e. G
14	1, 2, 3, 10, 13	strss 15910	. . 3 |- ( +g ` R ) = ( +g ` G )
15	9, 14	grpprop 17438	. 2 |- ( R e. Grp <-> G e. Grp )
16	15	bicomi 214	1 |- ( G e. Grp <-> R e. Grp )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {cpr 4179   <.cop 4183   Fun wfun 5882   ` cfv 5888   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by: (None)