Theorem isgrp 13138

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 5558	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		isgrp.b	. . . 4 |- B = ( Base ` G )
3	1, 2	eqtr4di 2247	. . 3 |- ( g = G -> ( Base ` g ) = B )
4		fveq2 5558	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
5		isgrp.p	. . . . . . 7 |- .+ = ( +g ` G )
6	4, 5	eqtr4di 2247	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
7	6	oveqd 5939	. . . . 5 |- ( g = G -> ( m ( +g ` g ) a ) = ( m .+ a ) )
8		fveq2 5558	. . . . . 6 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
9		isgrp.z	. . . . . 6 |- .0. = ( 0g ` G )
10	8, 9	eqtr4di 2247	. . . . 5 |- ( g = G -> ( 0g ` g ) = .0. )
11	7, 10	eqeq12d 2211	. . . 4 |- ( g = G -> ( ( m ( +g ` g ) a ) = ( 0g ` g ) <-> ( m .+ a ) = .0. ) )
12	3, 11	rexeqbidv 2710	. . 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 2709	. 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 13135	. 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 2923	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 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057   Grpcgrp 13132

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-grp 13135

This theorem is referenced by:  grpmnd  13139  grpinvex  13142  grppropd  13149  isgrpd2e  13152  grp1  13238  ghmgrp  13248