Theorem grpidrcan 13850

Description: If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem grpidrcan

Step	Hyp	Ref	Expression
1		grpidrcan.b	. . . . 5 |- B = ( Base ` G )
2		grpidrcan.p	. . . . 5 |- .+ = ( +g ` G )
3		grpidrcan.o	. . . . 5 |- .0. = ( 0g ` G )
4	1, 2, 3	grprid 13817	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ .0. ) = X )
5	4	3adant3 1048	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( X .+ .0. ) = X )
6	5	eqeq2d 2250	. 2 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) = ( X .+ .0. ) <-> ( X .+ Z ) = X ) )
7		simp1 1028	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> G e. Grp )
8		simp3 1030	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> Z e. B )
9	1, 3	grpidcl 13814	. . . 4 |- ( G e. Grp -> .0. e. B )
10	9	3ad2ant1 1049	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> .0. e. B )
11		simp2 1029	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> X e. B )
12	1, 2	grplcan 13847	. . 3 |- ( ( G e. Grp /\ ( Z e. B /\ .0. e. B /\ X e. B ) ) -> ( ( X .+ Z ) = ( X .+ .0. ) <-> Z = .0. ) )
13	7, 8, 10, 11, 12	syl13anc 1280	. 2 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) = ( X .+ .0. ) <-> Z = .0. ) )
14	6, 13	bitr3d 190	1 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) = X <-> Z = .0. ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6078   Basecbs 13333   +g cplusg 13411   0gc0g 13590   Grpcgrp 13785

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 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-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8263  ax-resscn 8264  ax-1re 8266  ax-addrcl 8269

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-inn 9287  df-2 9345  df-ndx 13336  df-slot 13337  df-base 13339  df-plusg 13424  df-0g 13592  df-mgm 13656  df-sgrp 13697  df-mnd 13710  df-grp 13788  df-minusg 13789

This theorem is referenced by: (None)