Theorem grpidinv 13131

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 2193	. . 3 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	grpidcl 13101	. 2 |- ( G e. Grp -> ( 0g ` G ) e. B )
4		oveq1 5925	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( u .+ x ) = ( ( 0g ` G ) .+ x ) )
5	4	eqeq1d 2202	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( u .+ x ) = x <-> ( ( 0g ` G ) .+ x ) = x ) )
6		oveq2 5926	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( x .+ u ) = ( x .+ ( 0g ` G ) ) )
7	6	eqeq1d 2202	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( x .+ u ) = x <-> ( x .+ ( 0g ` G ) ) = x ) )
8	5, 7	anbi12d 473	. . . . 5 |- ( u = ( 0g ` G ) -> ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) ) )
9		eqeq2 2203	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( ( y .+ x ) = u <-> ( y .+ x ) = ( 0g ` G ) ) )
10		eqeq2 2203	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( ( x .+ y ) = u <-> ( x .+ y ) = ( 0g ` G ) ) )
11	9, 10	anbi12d 473	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( ( y .+ x ) = u /\ ( x .+ y ) = u ) <-> ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) )
12	11	rexbidv 2495	. . . . 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 473	. . . 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 2494	. . 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 13130	. . 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 2567	. 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 2870	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 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076

This theorem is referenced by: (None)