Theorem grpidinv2 13133

Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised 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
grpidinv2	|- ( ( G e. Grp /\ A e. B ) -> ( ( ( .0. .+ A ) = A /\ ( A .+ .0. ) = A ) /\ E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )

Distinct variable groups:   y, B   y, G   y, .+   y, .0.   y, A

Proof of Theorem grpidinv2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grplrinv.b	. . 3 |- B = ( Base ` G )
2		grplrinv.p	. . 3 |- .+ = ( +g ` G )
3		grplrinv.i	. . 3 |- .0. = ( 0g ` G )
4	1, 2, 3	grplid 13106	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( .0. .+ A ) = A )
5	1, 2, 3	grprid 13107	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( A .+ .0. ) = A )
6	1, 2, 3	grplrinv 13132	. . 3 |- ( G e. Grp -> A. z e. B E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) )
7		oveq2 5927	. . . . . . 7 |- ( z = A -> ( y .+ z ) = ( y .+ A ) )
8	7	eqeq1d 2202	. . . . . 6 |- ( z = A -> ( ( y .+ z ) = .0. <-> ( y .+ A ) = .0. ) )
9		oveq1 5926	. . . . . . 7 |- ( z = A -> ( z .+ y ) = ( A .+ y ) )
10	9	eqeq1d 2202	. . . . . 6 |- ( z = A -> ( ( z .+ y ) = .0. <-> ( A .+ y ) = .0. ) )
11	8, 10	anbi12d 473	. . . . 5 |- ( z = A -> ( ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) <-> ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
12	11	rexbidv 2495	. . . 4 |- ( z = A -> ( E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) <-> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
13	12	rspcv 2861	. . 3 |- ( A e. B -> ( A. z e. B E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) -> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
14	6, 13	mpan9 281	. 2 |- ( ( G e. Grp /\ A e. B ) -> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) )
15	4, 5, 14	jca31 309	1 |- ( ( G e. Grp /\ A e. B ) -> ( ( ( .0. .+ A ) = A /\ ( A .+ .0. ) = A ) /\ E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Grpcgrp 13075

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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079

This theorem is referenced by:  grpidinv  13134