Theorem grpidd 12807

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 2177	. 2 |- ( Base ` G ) = ( Base ` G )
2		eqid 2177	. 2 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2177	. 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 2256	. 2 |- ( ph -> .0. e. ( Base ` G ) )
7	5	eleq2d 2247	. . . 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 5894	. . . 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 2212	. . 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 5894	. . . 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 2212	. . 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 12804	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   0gc0g 12710

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-0g 12712

This theorem is referenced by:  ress0g  12849  mnd1id  12853  isgrpde  12903