Theorem grpinvid2 13125

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 5925	. . . 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 13122	. . . . 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 2228	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( Y .+ X ) = .0. )
11	3, 6	grpinvcl 13120	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
12	3, 4, 5	grplid 13103	. . . . . . 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 2199	. . . 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 5925	. . . 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 13081	. . . . . . 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 13123	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
27	26	oveq2d 5934	. . . . . 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 13104	. . . . . 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 2230	. . . 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 2232	. 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 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072   invgcminusg 13073

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:  grpinvcnv  13140  grpsubeq0  13158  rngmneg2  13444  ringnegr  13548