Theorem isgrpd2e 13427

Description: Deduce a group from its properties. In this version of isgrpd2 13428, 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 2580	. . 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 5974	. . . . . 6 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
7		isgrpd2.z	. . . . . 6 |- ( ph -> .0. = ( 0g ` G ) )
8	6, 7	eqeq12d 2221	. . . . 5 |- ( ph -> ( ( y .+ x ) = .0. <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9	4, 8	rexeqbidv 2720	. . . 4 |- ( ph -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
10	4, 9	raleqbidv 2719	. . 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 2206	. . 3 |- ( Base ` G ) = ( Base ` G )
13		eqid 2206	. . 3 |- ( +g ` G ) = ( +g ` G )
14		eqid 2206	. . 3 |- ( 0g ` G ) = ( 0g ` G )
15	12, 13, 14	isgrp 13413	. 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 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Mndcmnd 13323   Grpcgrp 13407

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 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288  df-ov 5960  df-grp 13410

This theorem is referenced by:  isgrpd2  13428  isgrpde  13429