Theorem grpidlcan 13138

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 13103	. . . 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 2205	. 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 13101	. . . 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 13109	. . 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 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075

This theorem is referenced by:  grpidssd  13148