Theorem grpidrcan 13197

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 13164	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ .0. ) = X )
5	4	3adant3 1019	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( X .+ .0. ) = X )
6	5	eqeq2d 2208	. 2 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) = ( X .+ .0. ) <-> ( X .+ Z ) = X ) )
7		simp1 999	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> G e. Grp )
8		simp3 1001	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> Z e. B )
9	1, 3	grpidcl 13161	. . . 4 |- ( G e. Grp -> .0. e. B )
10	9	3ad2ant1 1020	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> .0. e. B )
11		simp2 1000	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> X e. B )
12	1, 2	grplcan 13194	. . 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 1251	. 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 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136

This theorem is referenced by: (None)