Theorem isgrpd2 13549

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 2229, but we make an exception for theorems such as isgrpd2 13549 and ismndd 13465 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 6007	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
8	7	eqeq1d 2238	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
9	8	rspcev 2907	. . 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 13548	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Mndcmnd 13444   Grpcgrp 13528

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003  df-grp 13531

This theorem is referenced by:  prdsgrpd  13637