Theorem grpidd2 13569

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13551. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
grpidd2.b	|- ( ph -> B = ( Base ` G ) )
grpidd2.p	|- ( ph -> .+ = ( +g ` G ) )
grpidd2.z	|- ( ph -> .0. e. B )
grpidd2.i	|- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
grpidd2.j	|- ( ph -> G e. Grp )

Assertion

Ref	Expression
grpidd2	|- ( ph -> .0. = ( 0g ` G ) )

Distinct variable groups:   x, B   x, .+   ph, x   x, .0.

Allowed substitution hint:   G( x)

Proof of Theorem grpidd2

Step	Hyp	Ref	Expression
1		grpidd2.p	. . . . 5 |- ( ph -> .+ = ( +g ` G ) )
2	1	oveqd 6017	. . . 4 |- ( ph -> ( .0. .+ .0. ) = ( .0. ( +g ` G ) .0. ) )
3		oveq2 6008	. . . . . 6 |- ( x = .0. -> ( .0. .+ x ) = ( .0. .+ .0. ) )
4		id 19	. . . . . 6 |- ( x = .0. -> x = .0. )
5	3, 4	eqeq12d 2244	. . . . 5 |- ( x = .0. -> ( ( .0. .+ x ) = x <-> ( .0. .+ .0. ) = .0. ) )
6		grpidd2.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
7	6	ralrimiva 2603	. . . . 5 |- ( ph -> A. x e. B ( .0. .+ x ) = x )
8		grpidd2.z	. . . . 5 |- ( ph -> .0. e. B )
9	5, 7, 8	rspcdva 2912	. . . 4 |- ( ph -> ( .0. .+ .0. ) = .0. )
10	2, 9	eqtr3d 2264	. . 3 |- ( ph -> ( .0. ( +g ` G ) .0. ) = .0. )
11		grpidd2.j	. . . 4 |- ( ph -> G e. Grp )
12		grpidd2.b	. . . . 5 |- ( ph -> B = ( Base ` G ) )
13	8, 12	eleqtrd 2308	. . . 4 |- ( ph -> .0. e. ( Base ` G ) )
14		eqid 2229	. . . . 5 |- ( Base ` G ) = ( Base ` G )
15		eqid 2229	. . . . 5 |- ( +g ` G ) = ( +g ` G )
16		eqid 2229	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
17	14, 15, 16	grpid 13567	. . . 4 |- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
18	11, 13, 17	syl2anc 411	. . 3 |- ( ph -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
19	10, 18	mpbid 147	. 2 |- ( ph -> ( 0g ` G ) = .0. )
20	19	eqcomd 2235	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Grpcgrp 13528

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531

This theorem is referenced by:  imasgrp2  13642