Theorem grpidd 17268

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 2622	. 2 |- ( Base ` G ) = ( Base ` G )
2		eqid 2622	. 2 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2622	. 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 2703	. 2 |- ( ph -> .0. e. ( Base ` G ) )
7	5	eleq2d 2687	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` G ) ) )
8	7	biimpar 502	. . 3 |- ( ( ph /\ x e. ( Base ` G ) ) -> x e. B )
9		grpidd.p	. . . . . 6 |- ( ph -> .+ = ( +g ` G ) )
10	9	adantr 481	. . . . 5 |- ( ( ph /\ x e. B ) -> .+ = ( +g ` G ) )
11	10	oveqd 6667	. . . 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 2658	. . 3 |- ( ( ph /\ x e. B ) -> ( .0. ( +g ` G ) x ) = x )
14	8, 13	syldan 487	. 2 |- ( ( ph /\ x e. ( Base ` G ) ) -> ( .0. ( +g ` G ) x ) = x )
15	10	oveqd 6667	. . . 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 2658	. . 3 |- ( ( ph /\ x e. B ) -> ( x ( +g ` G ) .0. ) = x )
18	8, 17	syldan 487	. 2 |- ( ( ph /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x )
19	1, 2, 3, 6, 14, 18	ismgmid2 17267	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102

This theorem is referenced by:  ress0g  17319  imasmnd2  17327  isgrpde  17443  xrs0  29675