Theorem isgrp 13453

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 5599	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		isgrp.b	. . . 4 |- B = ( Base ` G )
3	1, 2	eqtr4di 2258	. . 3 |- ( g = G -> ( Base ` g ) = B )
4		fveq2 5599	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
5		isgrp.p	. . . . . . 7 |- .+ = ( +g ` G )
6	4, 5	eqtr4di 2258	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
7	6	oveqd 5984	. . . . 5 |- ( g = G -> ( m ( +g ` g ) a ) = ( m .+ a ) )
8		fveq2 5599	. . . . . 6 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
9		isgrp.z	. . . . . 6 |- .0. = ( 0g ` G )
10	8, 9	eqtr4di 2258	. . . . 5 |- ( g = G -> ( 0g ` g ) = .0. )
11	7, 10	eqeq12d 2222	. . . 4 |- ( g = G -> ( ( m ( +g ` g ) a ) = ( 0g ` g ) <-> ( m .+ a ) = .0. ) )
12	3, 11	rexeqbidv 2722	. . 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 2721	. 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 13450	. 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 2939	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 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Mndcmnd 13363   Grpcgrp 13447

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970  df-grp 13450

This theorem is referenced by:  grpmnd  13454  grpinvex  13457  grppropd  13464  isgrpd2e  13467  grp1  13553  ghmgrp  13569