Theorem isgrpd2e 17441

Description: Deduce a group from its properties. In this version of isgrpd2 17442, 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 2966	. . 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 6667	. . . . . 6 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
7		isgrpd2.z	. . . . . 6 |- ( ph -> .0. = ( 0g ` G ) )
8	6, 7	eqeq12d 2637	. . . . 5 |- ( ph -> ( ( y .+ x ) = .0. <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	4, 8	rexeqbidv 3153	. . . 4 |- ( ph -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
10	4, 9	raleqbidv 3152	. . 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 222	. 2 |- ( ph -> A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) )
12		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
13		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
14		eqid 2622	. . 3 |- ( 0g ` G ) = ( 0g ` G )
15	12, 13, 14	isgrp 17428	. 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 698	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-grp 17425

This theorem is referenced by:  isgrpd2  17442  isgrpde  17443