Theorem grpidinv 13841

Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem grpidinv

Step	Hyp	Ref	Expression
1		grpidinv.b	. . 3 |- B = ( Base ` G )
2		eqid 2238	. . 3 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	grpidcl 13811	. 2 |- ( G e. Grp -> ( 0g ` G ) e. B )
4		oveq1 6082	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( u .+ x ) = ( ( 0g ` G ) .+ x ) )
5	4	eqeq1d 2247	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( u .+ x ) = x <-> ( ( 0g ` G ) .+ x ) = x ) )
6		oveq2 6083	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( x .+ u ) = ( x .+ ( 0g ` G ) ) )
7	6	eqeq1d 2247	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( x .+ u ) = x <-> ( x .+ ( 0g ` G ) ) = x ) )
8	5, 7	anbi12d 477	. . . . 5 |- ( u = ( 0g ` G ) -> ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) ) )
9		eqeq2 2248	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( ( y .+ x ) = u <-> ( y .+ x ) = ( 0g ` G ) ) )
10		eqeq2 2248	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( ( x .+ y ) = u <-> ( x .+ y ) = ( 0g ` G ) ) )
11	9, 10	anbi12d 477	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( ( y .+ x ) = u /\ ( x .+ y ) = u ) <-> ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) )
12	11	rexbidv 2551	. . . . 5 |- ( u = ( 0g ` G ) -> ( E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) <-> E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) )
13	8, 12	anbi12d 477	. . . 4 |- ( u = ( 0g ` G ) -> ( ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) <-> ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) )
14	13	ralbidv 2550	. . 3 |- ( u = ( 0g ` G ) -> ( A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) <-> A. x e. B ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) )
15	14	adantl 277	. 2 |- ( ( G e. Grp /\ u = ( 0g ` G ) ) -> ( A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) <-> A. x e. B ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) )
16		grpidinv.p	. . . 4 |- .+ = ( +g ` G )
17	1, 16, 2	grpidinv2 13840	. . 3 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) )
18	17	ralrimiva 2623	. 2 |- ( G e. Grp -> A. x e. B ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) )
19	3, 15, 18	rspcedvd 2935	1 |- ( G e. Grp -> E. u e. B A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786

This theorem is referenced by: (None)