Theorem grplrinv 13703

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 13694	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
4		oveq1 6035	. . . . . 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 6036	. . . . . 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 13696	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( invg ` G ) ` x ) .+ x ) = .0. )
13	1, 10, 11, 2	grprinv 13697	. . . 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 2917	. 2 |- ( ( G e. Grp /\ x e. B ) -> E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )
16	15	ralrimiva 2606	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 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   0gc0g 13402   Grpcgrp 13646   invgcminusg 13647

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650

This theorem is referenced by:  grpidinv2  13704