Theorem isgrpd2 13324

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 2204, but we make an exception for theorems such as isgrpd2 13324 and ismndd 13240 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 5950	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
8	7	eqeq1d 2213	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
9	8	rspcev 2876	. . 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 13323	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   E.wrex 2484   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   0gc0g 13059   Mndcmnd 13219   Grpcgrp 13303

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-ov 5946  df-grp 13306

This theorem is referenced by:  prdsgrpd  13412