Theorem isgrpd2 13734

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 2232, but we make an exception for theorems such as isgrpd2 13734 and ismndd 13650 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 6057	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
8	7	eqeq1d 2241	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
9	8	rspcev 2921	. . 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 13733	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   E.wrex 2521   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Mndcmnd 13629   Grpcgrp 13713

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053  df-grp 13716

This theorem is referenced by:  prdsgrpd  13822