Theorem grpidlcan 13316

Description: If left 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
grpidlcan	|- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( Z .+ X ) = X <-> Z = .0. ) )

Proof of Theorem grpidlcan

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	grplid 13281	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )
5	4	3adant3 1019	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( .0. .+ X ) = X )
6	5	eqeq2d 2216	. 2 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( Z .+ X ) = ( .0. .+ X ) <-> ( Z .+ X ) = 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 13279	. . . 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	grprcan 13287	. . 3 |- ( ( G e. Grp /\ ( Z e. B /\ .0. e. B /\ X e. B ) ) -> ( ( Z .+ X ) = ( .0. .+ X ) <-> Z = .0. ) )
13	7, 8, 10, 11, 12	syl13anc 1251	. 2 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( Z .+ X ) = ( .0. .+ X ) <-> Z = .0. ) )
14	6, 13	bitr3d 190	1 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( Z .+ X ) = X <-> Z = .0. ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   ` cfv 5268  (class class class)co 5934   Basecbs 12751   +g cplusg 12828   0gc0g 13006   Grpcgrp 13250

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fn 5271  df-fv 5276  df-riota 5889  df-ov 5937  df-inn 9019  df-2 9077  df-ndx 12754  df-slot 12755  df-base 12757  df-plusg 12841  df-0g 13008  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-grp 13253

This theorem is referenced by:  grpidssd  13326