Theorem grpo2grp 23853

Description: Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grp2grp.a	|- ( Base ` K ) = ran .+
grp2grp.p	|- ( +g ` K ) = .+
grp2grp.g	|- .+ e. GrpOp

Assertion

Ref	Expression
grpo2grp	|- K e. Grp

Proof of Theorem grpo2grp

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grp2grp.a	. . 3 |- ( Base ` K ) = ran .+
2	1	eqcomi 2463	. 2 |- ran .+ = ( Base ` K )
3		grp2grp.p	. . 3 |- ( +g ` K ) = .+
4	3	eqcomi 2463	. 2 |- .+ = ( +g ` K )
5		grp2grp.g	. . 3 |- .+ e. GrpOp
6		eqid 2451	. . . 4 |- ran .+ = ran .+
7	6	grpocl 23819	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ /\ b e. ran .+ ) -> ( a .+ b ) e. ran .+ )
8	5, 7	mp3an1 1302	. 2 |- ( ( a e. ran .+ /\ b e. ran .+ ) -> ( a .+ b ) e. ran .+ )
9	6	grpoass 23822	. . 3 |- ( ( .+ e. GrpOp /\ ( a e. ran .+ /\ b e. ran .+ /\ c e. ran .+ ) ) -> ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) )
10	5, 9	mpan 670	. 2 |- ( ( a e. ran .+ /\ b e. ran .+ /\ c e. ran .+ ) -> ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) )
11		eqid 2451	. . . 4 |- (GId ` .+ ) = (GId ` .+ )
12	6, 11	grpoidcl 23836	. . 3 |- ( .+ e. GrpOp -> (GId ` .+ ) e. ran .+ )
13	5, 12	ax-mp 5	. 2 |- (GId ` .+ ) e. ran .+
14	6, 11	grpolid 23838	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ ) -> ( (GId ` .+ ) .+ a ) = a )
15	5, 14	mpan 670	. 2 |- ( a e. ran .+ -> ( (GId ` .+ ) .+ a ) = a )
16		eqid 2451	. . . 4 |- ( inv ` .+ ) = ( inv ` .+ )
17	6, 16	grpoinvcl 23845	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ ) -> ( ( inv ` .+ ) ` a ) e. ran .+ )
18	5, 17	mpan 670	. 2 |- ( a e. ran .+ -> ( ( inv ` .+ ) ` a ) e. ran .+ )
19	6, 11, 16	grpolinv 23847	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ ) -> ( ( ( inv ` .+ ) ` a ) .+ a ) = (GId ` .+ ) )
20	5, 19	mpan 670	. 2 |- ( a e. ran .+ -> ( ( ( inv ` .+ ) ` a ) .+ a ) = (GId ` .+ ) )
21	2, 4, 8, 10, 13, 15, 18, 20	isgrpi 15663	1 |- K e. Grp

Syntax hints:   /\ w3a 965   = wceq 1370   e. wcel 1758   ran crn 4936   ` cfv 5513  (class class class)co 6187   Basecbs 14273   +g cplusg 14337   Grpcgrp 15509   GrpOpcgr 23805  GIdcgi 23806   invcgn 23807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-0g 14479  df-mnd 15514  df-grp 15644  df-grpo 23810  df-gid 23811  df-ginv 23812

This theorem is referenced by: (None)