Theorem grpidinv2 13190

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 13163	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( .0. .+ A ) = A )
5	1, 2, 3	grprid 13164	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( A .+ .0. ) = A )
6	1, 2, 3	grplrinv 13189	. . 3 |- ( G e. Grp -> A. z e. B E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) )
7		oveq2 5930	. . . . . . 7 |- ( z = A -> ( y .+ z ) = ( y .+ A ) )
8	7	eqeq1d 2205	. . . . . 6 |- ( z = A -> ( ( y .+ z ) = .0. <-> ( y .+ A ) = .0. ) )
9		oveq1 5929	. . . . . . 7 |- ( z = A -> ( z .+ y ) = ( A .+ y ) )
10	9	eqeq1d 2205	. . . . . 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 2498	. . . 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 2864	. . 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 2167   A.wral 2475   E.wrex 2476   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136

This theorem is referenced by:  grpidinv  13191