Theorem grpidlcan 13473

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 13438	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )
5	4	3adant3 1020	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( .0. .+ X ) = X )
6	5	eqeq2d 2218	. 2 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( Z .+ X ) = ( .0. .+ X ) <-> ( Z .+ X ) = X ) )
7		simp1 1000	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> G e. Grp )
8		simp3 1002	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> Z e. B )
9	1, 3	grpidcl 13436	. . . 4 |- ( G e. Grp -> .0. e. B )
10	9	3ad2ant1 1021	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> .0. e. B )
11		simp2 1001	. . 3 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> X e. B )
12	1, 2	grprcan 13444	. . 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 1252	. 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 981   = wceq 1373   e. wcel 2177   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Grpcgrp 13407

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288  df-riota 5912  df-ov 5960  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410

This theorem is referenced by:  grpidssd  13483