Theorem isgrp 13309

Description: The predicate "is a group". (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
isgrp.b	|- B = ( Base ` G )
isgrp.p	|- .+ = ( +g ` G )
isgrp.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
isgrp	|- ( G e. Grp <-> ( G e. Mnd /\ A. a e. B E. m e. B ( m .+ a ) = .0. ) )

Distinct variable groups:   m, a, B   G, a, m

Allowed substitution hints:   .+ ( m, a)   .0. ( m, a)

Proof of Theorem isgrp

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5575	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		isgrp.b	. . . 4 |- B = ( Base ` G )
3	1, 2	eqtr4di 2255	. . 3 |- ( g = G -> ( Base ` g ) = B )
4		fveq2 5575	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
5		isgrp.p	. . . . . . 7 |- .+ = ( +g ` G )
6	4, 5	eqtr4di 2255	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
7	6	oveqd 5960	. . . . 5 |- ( g = G -> ( m ( +g ` g ) a ) = ( m .+ a ) )
8		fveq2 5575	. . . . . 6 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
9		isgrp.z	. . . . . 6 |- .0. = ( 0g ` G )
10	8, 9	eqtr4di 2255	. . . . 5 |- ( g = G -> ( 0g ` g ) = .0. )
11	7, 10	eqeq12d 2219	. . . 4 |- ( g = G -> ( ( m ( +g ` g ) a ) = ( 0g ` g ) <-> ( m .+ a ) = .0. ) )
12	3, 11	rexeqbidv 2718	. . 3 |- ( g = G -> ( E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) <-> E. m e. B ( m .+ a ) = .0. ) )
13	3, 12	raleqbidv 2717	. 2 |- ( g = G -> ( A. a e. ( Base ` g ) E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) <-> A. a e. B E. m e. B ( m .+ a ) = .0. ) )
14		df-grp 13306	. 2 |- Grp = { g e. Mnd | A. a e. ( Base ` g ) E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) }
15	13, 14	elrab2 2931	1 |- ( G e. Grp <-> ( G e. Mnd /\ A. a e. B E. m e. B ( m .+ a ) = .0. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   0gc0g 13059   Mndcmnd 13219   Grpcgrp 13303

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-ov 5946  df-grp 13306

This theorem is referenced by:  grpmnd  13310  grpinvex  13313  grppropd  13320  isgrpd2e  13323  grp1  13409  ghmgrp  13425