Theorem isgrpd2e 12726

Description: Deduce a group from its properties. In this version of isgrpd2 12727, 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 2543	. . 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 5870	. . . . . 6 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
7		isgrpd2.z	. . . . . 6 |- ( ph -> .0. = ( 0g ` G ) )
8	6, 7	eqeq12d 2185	. . . . 5 |- ( ph -> ( ( y .+ x ) = .0. <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	4, 8	rexeqbidv 2678	. . . 4 |- ( ph -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
10	4, 9	raleqbidv 2677	. . 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 146	. 2 |- ( ph -> A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) )
12		eqid 2170	. . 3 |- ( Base ` G ) = ( Base ` G )
13		eqid 2170	. . 3 |- ( +g ` G ) = ( +g ` G )
14		eqid 2170	. . 3 |- ( 0g ` G ) = ( 0g ` G )
15	12, 13, 14	isgrp 12714	. 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 415	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   ` cfv 5198  (class class class)co 5853   Basecbs 12416   +g cplusg 12480   0gc0g 12596   Mndcmnd 12652   Grpcgrp 12708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856  df-grp 12711

This theorem is referenced by:  isgrpd2  12727  isgrpde  12728