Theorem isgrp 13588

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 5639	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		isgrp.b	. . . 4 |- B = ( Base ` G )
3	1, 2	eqtr4di 2282	. . 3 |- ( g = G -> ( Base ` g ) = B )
4		fveq2 5639	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
5		isgrp.p	. . . . . . 7 |- .+ = ( +g ` G )
6	4, 5	eqtr4di 2282	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
7	6	oveqd 6034	. . . . 5 |- ( g = G -> ( m ( +g ` g ) a ) = ( m .+ a ) )
8		fveq2 5639	. . . . . 6 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
9		isgrp.z	. . . . . 6 |- .0. = ( 0g ` G )
10	8, 9	eqtr4di 2282	. . . . 5 |- ( g = G -> ( 0g ` g ) = .0. )
11	7, 10	eqeq12d 2246	. . . 4 |- ( g = G -> ( ( m ( +g ` g ) a ) = ( 0g ` g ) <-> ( m .+ a ) = .0. ) )
12	3, 11	rexeqbidv 2747	. . 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 2746	. 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 13585	. 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 2965	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 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Mndcmnd 13498   Grpcgrp 13582

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-grp 13585

This theorem is referenced by:  grpmnd  13589  grpinvex  13592  grppropd  13599  isgrpd2e  13602  grp1  13688  ghmgrp  13704