Theorem grplrinv 13585

Description: In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021.)

Hypotheses

Ref	Expression
grplrinv.b	|- B = ( Base ` G )
grplrinv.p	|- .+ = ( +g ` G )
grplrinv.i	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grplrinv	|- ( G e. Grp -> A. x e. B E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )

Distinct variable groups:   y, B   x, G, y   y, .+   y, .0.

Allowed substitution hints:   B( x)   .+ ( x)   .0. ( x)

Proof of Theorem grplrinv

Step	Hyp	Ref	Expression
1		grplrinv.b	. . . 4 |- B = ( Base ` G )
2		eqid 2229	. . . 4 |- ( invg ` G ) = ( invg ` G )
3	1, 2	grpinvcl 13576	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
4		oveq1 6007	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( y .+ x ) = ( ( ( invg ` G ) ` x ) .+ x ) )
5	4	eqeq1d 2238	. . . . 5 |- ( y = ( ( invg ` G ) ` x ) -> ( ( y .+ x ) = .0. <-> ( ( ( invg ` G ) ` x ) .+ x ) = .0. ) )
6		oveq2 6008	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( x .+ y ) = ( x .+ ( ( invg ` G ) ` x ) ) )
7	6	eqeq1d 2238	. . . . 5 |- ( y = ( ( invg ` G ) ` x ) -> ( ( x .+ y ) = .0. <-> ( x .+ ( ( invg ` G ) ` x ) ) = .0. ) )
8	5, 7	anbi12d 473	. . . 4 |- ( y = ( ( invg ` G ) ` x ) -> ( ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) <-> ( ( ( ( invg ` G ) ` x ) .+ x ) = .0. /\ ( x .+ ( ( invg ` G ) ` x ) ) = .0. ) ) )
9	8	adantl 277	. . 3 |- ( ( ( G e. Grp /\ x e. B ) /\ y = ( ( invg ` G ) ` x ) ) -> ( ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) <-> ( ( ( ( invg ` G ) ` x ) .+ x ) = .0. /\ ( x .+ ( ( invg ` G ) ` x ) ) = .0. ) ) )
10		grplrinv.p	. . . . 5 |- .+ = ( +g ` G )
11		grplrinv.i	. . . . 5 |- .0. = ( 0g ` G )
12	1, 10, 11, 2	grplinv 13578	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( invg ` G ) ` x ) .+ x ) = .0. )
13	1, 10, 11, 2	grprinv 13579	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( x .+ ( ( invg ` G ) ` x ) ) = .0. )
14	12, 13	jca 306	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( ( invg ` G ) ` x ) .+ x ) = .0. /\ ( x .+ ( ( invg ` G ) ` x ) ) = .0. ) )
15	3, 9, 14	rspcedvd 2913	. 2 |- ( ( G e. Grp /\ x e. B ) -> E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )
16	15	ralrimiva 2603	1 |- ( G e. Grp -> A. x e. B E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Grpcgrp 13528   invgcminusg 13529

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532

This theorem is referenced by:  grpidinv2  13586