Theorem grplrinv 13639

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 2231	. . . 4 |- ( invg ` G ) = ( invg ` G )
3	1, 2	grpinvcl 13630	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
4		oveq1 6024	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( y .+ x ) = ( ( ( invg ` G ) ` x ) .+ x ) )
5	4	eqeq1d 2240	. . . . 5 |- ( y = ( ( invg ` G ) ` x ) -> ( ( y .+ x ) = .0. <-> ( ( ( invg ` G ) ` x ) .+ x ) = .0. ) )
6		oveq2 6025	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( x .+ y ) = ( x .+ ( ( invg ` G ) ` x ) ) )
7	6	eqeq1d 2240	. . . . 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 13632	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( invg ` G ) ` x ) .+ x ) = .0. )
13	1, 10, 11, 2	grprinv 13633	. . . 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 2916	. 2 |- ( ( G e. Grp /\ x e. B ) -> E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )
16	15	ralrimiva 2605	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 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586

This theorem is referenced by:  grpidinv2  13640