Theorem grplrinv 13389

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 2205	. . . 4 |- ( invg ` G ) = ( invg ` G )
3	1, 2	grpinvcl 13380	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
4		oveq1 5951	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( y .+ x ) = ( ( ( invg ` G ) ` x ) .+ x ) )
5	4	eqeq1d 2214	. . . . 5 |- ( y = ( ( invg ` G ) ` x ) -> ( ( y .+ x ) = .0. <-> ( ( ( invg ` G ) ` x ) .+ x ) = .0. ) )
6		oveq2 5952	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( x .+ y ) = ( x .+ ( ( invg ` G ) ` x ) ) )
7	6	eqeq1d 2214	. . . . 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 13382	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( invg ` G ) ` x ) .+ x ) = .0. )
13	1, 10, 11, 2	grprinv 13383	. . . 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 2883	. 2 |- ( ( G e. Grp /\ x e. B ) -> E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )
16	15	ralrimiva 2579	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 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088   Grpcgrp 13332   invgcminusg 13333

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336

This theorem is referenced by:  grpidinv2  13390