Theorem grplrinv 13504

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 2207	. . . 4 |- ( invg ` G ) = ( invg ` G )
3	1, 2	grpinvcl 13495	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
4		oveq1 5974	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( y .+ x ) = ( ( ( invg ` G ) ` x ) .+ x ) )
5	4	eqeq1d 2216	. . . . 5 |- ( y = ( ( invg ` G ) ` x ) -> ( ( y .+ x ) = .0. <-> ( ( ( invg ` G ) ` x ) .+ x ) = .0. ) )
6		oveq2 5975	. . . . . 6 |- ( y = ( ( invg ` G ) ` x ) -> ( x .+ y ) = ( x .+ ( ( invg ` G ) ` x ) ) )
7	6	eqeq1d 2216	. . . . 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 13497	. . . 4 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( invg ` G ) ` x ) .+ x ) = .0. )
13	1, 10, 11, 2	grprinv 13498	. . . 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 2890	. 2 |- ( ( G e. Grp /\ x e. B ) -> E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )
16	15	ralrimiva 2581	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 2178   A.wral 2486   E.wrex 2487   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447   invgcminusg 13448

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 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451

This theorem is referenced by:  grpidinv2  13505