Theorem grpid 13567

Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)

Hypotheses

Ref	Expression
grpinveu.b	|- B = ( Base ` G )
grpinveu.p	|- .+ = ( +g ` G )
grpinveu.o	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpid	|- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = X <-> .0. = X ) )

Proof of Theorem grpid

Step	Hyp	Ref	Expression
1		eqcom 2231	. 2 |- ( .0. = X <-> X = .0. )
2		grpinveu.b	. . . . . . 7 |- B = ( Base ` G )
3		grpinveu.o	. . . . . . 7 |- .0. = ( 0g ` G )
4	2, 3	grpidcl 13557	. . . . . 6 |- ( G e. Grp -> .0. e. B )
5		grpinveu.p	. . . . . . . 8 |- .+ = ( +g ` G )
6	2, 5	grprcan 13565	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ .0. e. B /\ X e. B ) ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) )
7	6	3exp2 1249	. . . . . 6 |- ( G e. Grp -> ( X e. B -> ( .0. e. B -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) ) ) )
8	4, 7	mpid 42	. . . . 5 |- ( G e. Grp -> ( X e. B -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) ) )
9	8	pm2.43d 50	. . . 4 |- ( G e. Grp -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) )
10	9	imp 124	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) )
11	2, 5, 3	grplid 13559	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )
12	11	eqeq2d 2241	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> ( X .+ X ) = X ) )
13	10, 12	bitr3d 190	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X = .0. <-> ( X .+ X ) = X ) )
14	1, 13	bitr2id 193	1 |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = X <-> .0. = X ) )

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:  isgrpid2  13568  grpidd2  13569  subg0  13712  qus0  13767  ghmid  13781  lmod0vid  14278  cnfld0  14529  psr0  14644