Theorem isgrpd2 13468

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 2207, but we make an exception for theorems such as isgrpd2 13468 and ismndd 13384 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 5974	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
8	7	eqeq1d 2216	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
9	8	rspcev 2884	. . 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 13467	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   E.wrex 2487   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Mndcmnd 13363   Grpcgrp 13447

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 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970  df-grp 13450

This theorem is referenced by:  prdsgrpd  13556