Theorem grpidd 13085

Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpidd.b	|- ( ph -> B = ( Base ` G ) )
grpidd.p	|- ( ph -> .+ = ( +g ` G ) )
grpidd.z	|- ( ph -> .0. e. B )
grpidd.i	|- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
grpidd.j	|- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )

Assertion

Ref	Expression
grpidd	|- ( ph -> .0. = ( 0g ` G ) )

Distinct variable groups:   x, G   ph, x   x, .0.

Allowed substitution hints:   B( x)   .+ ( x)

Proof of Theorem grpidd

Step	Hyp	Ref	Expression
1		eqid 2196	. 2 |- ( Base ` G ) = ( Base ` G )
2		eqid 2196	. 2 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2196	. 2 |- ( +g ` G ) = ( +g ` G )
4		grpidd.z	. . 3 |- ( ph -> .0. e. B )
5		grpidd.b	. . 3 |- ( ph -> B = ( Base ` G ) )
6	4, 5	eleqtrd 2275	. 2 |- ( ph -> .0. e. ( Base ` G ) )
7	5	eleq2d 2266	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` G ) ) )
8	7	biimpar 297	. . 3 |- ( ( ph /\ x e. ( Base ` G ) ) -> x e. B )
9		grpidd.p	. . . . . 6 |- ( ph -> .+ = ( +g ` G ) )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ x e. B ) -> .+ = ( +g ` G ) )
11	10	oveqd 5942	. . . 4 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = ( .0. ( +g ` G ) x ) )
12		grpidd.i	. . . 4 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
13	11, 12	eqtr3d 2231	. . 3 |- ( ( ph /\ x e. B ) -> ( .0. ( +g ` G ) x ) = x )
14	8, 13	syldan 282	. 2 |- ( ( ph /\ x e. ( Base ` G ) ) -> ( .0. ( +g ` G ) x ) = x )
15	10	oveqd 5942	. . . 4 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = ( x ( +g ` G ) .0. ) )
16		grpidd.j	. . . 4 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )
17	15, 16	eqtr3d 2231	. . 3 |- ( ( ph /\ x e. B ) -> ( x ( +g ` G ) .0. ) = x )
18	8, 17	syldan 282	. 2 |- ( ( ph /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x )
19	1, 2, 3, 6, 14, 18	ismgmid2 13082	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-0g 12960

This theorem is referenced by:  ress0g  13145  imasmnd2  13154  mnd1id  13158  isgrpde  13224