Theorem isgrpd2 12757

Description: Deduce a group from its properties. N (negative) is normally dependent on x i.e. read it as N ( x ). Note: normally we don't use a ph antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2175, but we make an exception for theorems such as isgrpd2 12757 and ismndd 12702 since theorems using them often rewrite the structure components. (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 )
isgrpd2.n	|- ( ( ph /\ x e. B ) -> N e. B )
isgrpd2.j	|- ( ( ph /\ x e. B ) -> ( N .+ x ) = .0. )

Assertion

Ref	Expression
isgrpd2	|- ( ph -> G e. Grp )

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

Allowed substitution hints:   N( x)   .0. ( x)

Proof of Theorem isgrpd2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpd2.b	. 2 |- ( ph -> B = ( Base ` G ) )
2		isgrpd2.p	. 2 |- ( ph -> .+ = ( +g ` G ) )
3		isgrpd2.z	. 2 |- ( ph -> .0. = ( 0g ` G ) )
4		isgrpd2.g	. 2 |- ( ph -> G e. Mnd )
5		isgrpd2.n	. . 3 |- ( ( ph /\ x e. B ) -> N e. B )
6		isgrpd2.j	. . 3 |- ( ( ph /\ x e. B ) -> ( N .+ x ) = .0. )
7		oveq1 5872	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
8	7	eqeq1d 2184	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
9	8	rspcev 2839	. . 3 |- ( ( N e. B /\ ( N .+ x ) = .0. ) -> E. y e. B ( y .+ x ) = .0. )
10	5, 6, 9	syl2anc 411	. 2 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )
11	1, 2, 3, 4, 10	isgrpd2e 12756	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   E.wrex 2454   ` cfv 5208  (class class class)co 5865   Basecbs 12427   +g cplusg 12491   0gc0g 12625   Mndcmnd 12681   Grpcgrp 12737

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868  df-grp 12740

This theorem is referenced by: (None)