Theorem grpo2grp 25437

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 2467	. 2 |- ran .+ = ( Base ` K )
3		grp2grp.p	. . 3 |- ( +g ` K ) = .+
4	3	eqcomi 2467	. 2 |- .+ = ( +g ` K )
5		grp2grp.g	. . 3 |- .+ e. GrpOp
6		eqid 2454	. . . 4 |- ran .+ = ran .+
7	6	grpocl 25403	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ /\ b e. ran .+ ) -> ( a .+ b ) e. ran .+ )
8	5, 7	mp3an1 1309	. 2 |- ( ( a e. ran .+ /\ b e. ran .+ ) -> ( a .+ b ) e. ran .+ )
9	6	grpoass 25406	. . 3 |- ( ( .+ e. GrpOp /\ ( a e. ran .+ /\ b e. ran .+ /\ c e. ran .+ ) ) -> ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) )
10	5, 9	mpan 668	. 2 |- ( ( a e. ran .+ /\ b e. ran .+ /\ c e. ran .+ ) -> ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) )
11		eqid 2454	. . . 4 |- (GId ` .+ ) = (GId ` .+ )
12	6, 11	grpoidcl 25420	. . 3 |- ( .+ e. GrpOp -> (GId ` .+ ) e. ran .+ )
13	5, 12	ax-mp 5	. 2 |- (GId ` .+ ) e. ran .+
14	6, 11	grpolid 25422	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ ) -> ( (GId ` .+ ) .+ a ) = a )
15	5, 14	mpan 668	. 2 |- ( a e. ran .+ -> ( (GId ` .+ ) .+ a ) = a )
16		eqid 2454	. . . 4 |- ( inv ` .+ ) = ( inv ` .+ )
17	6, 16	grpoinvcl 25429	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ ) -> ( ( inv ` .+ ) ` a ) e. ran .+ )
18	5, 17	mpan 668	. 2 |- ( a e. ran .+ -> ( ( inv ` .+ ) ` a ) e. ran .+ )
19	6, 11, 16	grpolinv 25431	. . 3 |- ( ( .+ e. GrpOp /\ a e. ran .+ ) -> ( ( ( inv ` .+ ) ` a ) .+ a ) = (GId ` .+ ) )
20	5, 19	mpan 668	. 2 |- ( a e. ran .+ -> ( ( ( inv ` .+ ) ` a ) .+ a ) = (GId ` .+ ) )
21	2, 4, 8, 10, 13, 15, 18, 20	isgrpi 16278	1 |- K e. Grp

Syntax hints:   /\ w3a 971   = wceq 1398   e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   Grpcgrp 16255   GrpOpcgr 25389  GIdcgi 25390   invcgn 25391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-grpo 25394  df-gid 25395  df-ginv 25396

This theorem is referenced by: (None)