Theorem grpidinv 12763

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 2171	. . 3 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	grpidcl 12738	. 2 |- ( G e. Grp -> ( 0g ` G ) e. B )
4		oveq1 5864	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( u .+ x ) = ( ( 0g ` G ) .+ x ) )
5	4	eqeq1d 2180	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( u .+ x ) = x <-> ( ( 0g ` G ) .+ x ) = x ) )
6		oveq2 5865	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( x .+ u ) = ( x .+ ( 0g ` G ) ) )
7	6	eqeq1d 2180	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( x .+ u ) = x <-> ( x .+ ( 0g ` G ) ) = x ) )
8	5, 7	anbi12d 471	. . . . 5 |- ( u = ( 0g ` G ) -> ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) ) )
9		eqeq2 2181	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( ( y .+ x ) = u <-> ( y .+ x ) = ( 0g ` G ) ) )
10		eqeq2 2181	. . . . . . 7 |- ( u = ( 0g ` G ) -> ( ( x .+ y ) = u <-> ( x .+ y ) = ( 0g ` G ) ) )
11	9, 10	anbi12d 471	. . . . . 6 |- ( u = ( 0g ` G ) -> ( ( ( y .+ x ) = u /\ ( x .+ y ) = u ) <-> ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) )
12	11	rexbidv 2472	. . . . 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 471	. . . 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 2471	. . 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 275	. 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 12762	. . 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 2544	. 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 2841	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 103   <-> wb 104   = wceq 1349   e. wcel 2142   A.wral 2449   E.wrex 2450   ` cfv 5200  (class class class)co 5857   Basecbs 12420   +g cplusg 12484   0gc0g 12600   Grpcgrp 12712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-cnex 7869  ax-resscn 7870  ax-1re 7872  ax-addrcl 7875

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-inn 8883  df-2 8941  df-ndx 12423  df-slot 12424  df-base 12426  df-plusg 12497  df-0g 12602  df-mgm 12614  df-sgrp 12647  df-mnd 12657  df-grp 12715  df-minusg 12716

This theorem is referenced by: (None)