Theorem grpidrcan 12975

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 12942	. . . 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 2201	. 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 12939	. . . 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 12972	. . 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 2160   ` cfv 5231  (class class class)co 5891   Basecbs 12480   +g cplusg 12555   0gc0g 12727   Grpcgrp 12911

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8938  df-2 8996  df-ndx 12483  df-slot 12484  df-base 12486  df-plusg 12568  df-0g 12729  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-minusg 12915

This theorem is referenced by: (None)