Theorem grpidd 13703

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 2238	. 2 |- ( Base ` G ) = ( Base ` G )
2		eqid 2238	. 2 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2238	. 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 2317	. 2 |- ( ph -> .0. e. ( Base ` G ) )
7	5	eleq2d 2308	. . . 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 6102	. . . 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 2273	. . 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 6102	. . . 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 2273	. . 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 13700	1 |- ( ph -> .0. = ( 0g ` G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-0g 13612

This theorem is used by:  ress0g  13756  imasmnd2  13759  mnd1id  13763  isgrpde  13827