Theorem isgrp 12883

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 5516	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		isgrp.b	. . . 4 |- B = ( Base ` G )
3	1, 2	eqtr4di 2228	. . 3 |- ( g = G -> ( Base ` g ) = B )
4		fveq2 5516	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
5		isgrp.p	. . . . . . 7 |- .+ = ( +g ` G )
6	4, 5	eqtr4di 2228	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
7	6	oveqd 5892	. . . . 5 |- ( g = G -> ( m ( +g ` g ) a ) = ( m .+ a ) )
8		fveq2 5516	. . . . . 6 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
9		isgrp.z	. . . . . 6 |- .0. = ( 0g ` G )
10	8, 9	eqtr4di 2228	. . . . 5 |- ( g = G -> ( 0g ` g ) = .0. )
11	7, 10	eqeq12d 2192	. . . 4 |- ( g = G -> ( ( m ( +g ` g ) a ) = ( 0g ` g ) <-> ( m .+ a ) = .0. ) )
12	3, 11	rexeqbidv 2686	. . 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 2685	. 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 12880	. 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 2897	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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   0gc0g 12705   Mndcmnd 12817   Grpcgrp 12877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-grp 12880

This theorem is referenced by:  grpmnd  12884  grpinvex  12887  grppropd  12893  isgrpd2e  12896  grp1  12976  ghmgrp  12982