Theorem isgrpd2e 13548

Description: Deduce a group from its properties. In this version of isgrpd2 13549, we don't assume there is an expression for the inverse of x. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
isgrpd2.b	|- ( ph -> B = ( Base ` G ) )
isgrpd2.p	|- ( ph -> .+ = ( +g ` G ) )
isgrpd2.z	|- ( ph -> .0. = ( 0g ` G ) )
isgrpd2.g	|- ( ph -> G e. Mnd )
isgrpd2e.n	|- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )

Assertion

Ref	Expression
isgrpd2e	|- ( ph -> G e. Grp )

Distinct variable groups:   x, y, .+   y, .0.   x, B, y   x, G, y   ph, x, y

Allowed substitution hint:   .0. ( x)

Proof of Theorem isgrpd2e

Step	Hyp	Ref	Expression
1		isgrpd2.g	. 2 |- ( ph -> G e. Mnd )
2		isgrpd2e.n	. . . 4 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )
3	2	ralrimiva 2603	. . 3 |- ( ph -> A. x e. B E. y e. B ( y .+ x ) = .0. )
4		isgrpd2.b	. . . 4 |- ( ph -> B = ( Base ` G ) )
5		isgrpd2.p	. . . . . . 7 |- ( ph -> .+ = ( +g ` G ) )
6	5	oveqd 6017	. . . . . 6 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
7		isgrpd2.z	. . . . . 6 |- ( ph -> .0. = ( 0g ` G ) )
8	6, 7	eqeq12d 2244	. . . . 5 |- ( ph -> ( ( y .+ x ) = .0. <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	4, 8	rexeqbidv 2745	. . . 4 |- ( ph -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
10	4, 9	raleqbidv 2744	. . 3 |- ( ph -> ( A. x e. B E. y e. B ( y .+ x ) = .0. <-> A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
11	3, 10	mpbid 147	. 2 |- ( ph -> A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) )
12		eqid 2229	. . 3 |- ( Base ` G ) = ( Base ` G )
13		eqid 2229	. . 3 |- ( +g ` G ) = ( +g ` G )
14		eqid 2229	. . 3 |- ( 0g ` G ) = ( 0g ` G )
15	12, 13, 14	isgrp 13534	. 2 |- ( G e. Grp <-> ( G e. Mnd /\ A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
16	1, 11, 15	sylanbrc 417	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Mndcmnd 13444   Grpcgrp 13528

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003  df-grp 13531

This theorem is referenced by:  isgrpd2  13549  isgrpde  13550