Theorem grpinvid2 13185

Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Hypotheses

Ref	Expression
grpinv.b	|- B = ( Base ` G )
grpinv.p	|- .+ = ( +g ` G )
grpinv.u	|- .0. = ( 0g ` G )
grpinv.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvid2	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) = Y <-> ( Y .+ X ) = .0. ) )

Proof of Theorem grpinvid2

Step	Hyp	Ref	Expression
1		oveq1 5929	. . . 4 |- ( ( N ` X ) = Y -> ( ( N ` X ) .+ X ) = ( Y .+ X ) )
2	1	adantl 277	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( ( N ` X ) .+ X ) = ( Y .+ X ) )
3		grpinv.b	. . . . . 6 |- B = ( Base ` G )
4		grpinv.p	. . . . . 6 |- .+ = ( +g ` G )
5		grpinv.u	. . . . . 6 |- .0. = ( 0g ` G )
6		grpinv.n	. . . . . 6 |- N = ( invg ` G )
7	3, 4, 5, 6	grplinv 13182	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
8	7	3adant3 1019	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) .+ X ) = .0. )
9	8	adantr 276	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( ( N ` X ) .+ X ) = .0. )
10	2, 9	eqtr3d 2231	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( Y .+ X ) = .0. )
11	3, 6	grpinvcl 13180	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
12	3, 4, 5	grplid 13163	. . . . . . 7 |- ( ( G e. Grp /\ ( N ` X ) e. B ) -> ( .0. .+ ( N ` X ) ) = ( N ` X ) )
13	11, 12	syldan 282	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ ( N ` X ) ) = ( N ` X ) )
14	13	3adant3 1019	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ ( N ` X ) ) = ( N ` X ) )
15	14	eqcomd 2202	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` X ) = ( .0. .+ ( N ` X ) ) )
16	15	adantr 276	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( Y .+ X ) = .0. ) -> ( N ` X ) = ( .0. .+ ( N ` X ) ) )
17		oveq1 5929	. . . 4 |- ( ( Y .+ X ) = .0. -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( .0. .+ ( N ` X ) ) )
18	17	adantl 277	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( Y .+ X ) = .0. ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( .0. .+ ( N ` X ) ) )
19		simprr 531	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
20		simprl 529	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> X e. B )
21	11	adantrr 479	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( N ` X ) e. B )
22	19, 20, 21	3jca 1179	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( Y e. B /\ X e. B /\ ( N ` X ) e. B ) )
23	3, 4	grpass 13141	. . . . . . 7 |- ( ( G e. Grp /\ ( Y e. B /\ X e. B /\ ( N ` X ) e. B ) ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( Y .+ ( X .+ ( N ` X ) ) ) )
24	22, 23	syldan 282	. . . . . 6 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( Y .+ ( X .+ ( N ` X ) ) ) )
25	24	3impb 1201	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( Y .+ ( X .+ ( N ` X ) ) ) )
26	3, 4, 5, 6	grprinv 13183	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
27	26	oveq2d 5938	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( Y .+ ( X .+ ( N ` X ) ) ) = ( Y .+ .0. ) )
28	27	3adant3 1019	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .+ ( X .+ ( N ` X ) ) ) = ( Y .+ .0. ) )
29	3, 4, 5	grprid 13164	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( Y .+ .0. ) = Y )
30	29	3adant2 1018	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .+ .0. ) = Y )
31	25, 28, 30	3eqtrd 2233	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = Y )
32	31	adantr 276	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( Y .+ X ) = .0. ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = Y )
33	16, 18, 32	3eqtr2d 2235	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( Y .+ X ) = .0. ) -> ( N ` X ) = Y )
34	10, 33	impbida 596	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) = Y <-> ( Y .+ X ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133

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:  grpinvcnv  13200  grpsubeq0  13218  rngmneg2  13504  ringnegr  13608