Theorem grpinvid1 13585

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
grpinvid1	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) = Y <-> ( X .+ Y ) = .0. ) )

Proof of Theorem grpinvid1

Step	Hyp	Ref	Expression
1		oveq2 6009	. . . 4 |- ( ( N ` X ) = Y -> ( X .+ ( N ` X ) ) = ( X .+ Y ) )
2	1	adantl 277	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( X .+ ( N ` X ) ) = ( X .+ Y ) )
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	grprinv 13584	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
8	7	3adant3 1041	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( N ` X ) ) = .0. )
9	8	adantr 276	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( X .+ ( N ` X ) ) = .0. )
10	2, 9	eqtr3d 2264	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( X .+ Y ) = .0. )
11		oveq2 6009	. . . 4 |- ( ( X .+ Y ) = .0. -> ( ( N ` X ) .+ ( X .+ Y ) ) = ( ( N ` X ) .+ .0. ) )
12	11	adantl 277	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( X .+ Y ) = .0. ) -> ( ( N ` X ) .+ ( X .+ Y ) ) = ( ( N ` X ) .+ .0. ) )
13	3, 4, 5, 6	grplinv 13583	. . . . . . . 8 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
14	13	oveq1d 6016	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
15	14	3adant3 1041	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
16	3, 6	grpinvcl 13581	. . . . . . . . . 10 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
17	16	adantrr 479	. . . . . . . . 9 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( N ` X ) e. B )
18		simprl 529	. . . . . . . . 9 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> X e. B )
19		simprr 531	. . . . . . . . 9 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
20	17, 18, 19	3jca 1201	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( N ` X ) e. B /\ X e. B /\ Y e. B ) )
21	3, 4	grpass 13542	. . . . . . . 8 |- ( ( G e. Grp /\ ( ( N ` X ) e. B /\ X e. B /\ Y e. B ) ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
22	20, 21	syldan 282	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
23	22	3impb 1223	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
24	15, 23	eqtr3d 2264	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
25	3, 4, 5	grplid 13564	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( .0. .+ Y ) = Y )
26	25	3adant2 1040	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = Y )
27	24, 26	eqtr3d 2264	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) .+ ( X .+ Y ) ) = Y )
28	27	adantr 276	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( X .+ Y ) = .0. ) -> ( ( N ` X ) .+ ( X .+ Y ) ) = Y )
29	3, 4, 5	grprid 13565	. . . . . 6 |- ( ( G e. Grp /\ ( N ` X ) e. B ) -> ( ( N ` X ) .+ .0. ) = ( N ` X ) )
30	16, 29	syldan 282	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ .0. ) = ( N ` X ) )
31	30	3adant3 1041	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) .+ .0. ) = ( N ` X ) )
32	31	adantr 276	. . 3 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( X .+ Y ) = .0. ) -> ( ( N ` X ) .+ .0. ) = ( N ` X ) )
33	12, 28, 32	3eqtr3rd 2271	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( X .+ Y ) = .0. ) -> ( N ` X ) = Y )
34	10, 33	impbida 598	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) = Y <-> ( X .+ Y ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Grpcgrp 13533   invgcminusg 13534

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537

This theorem is referenced by:  grpinvid  13593  grpinvcnv  13601  grpinvadd  13611  subginv  13718  qusinv  13773  ghminv  13787  rngmneg1  13910  ringnegl  14014  lmodindp1  14392  cnfldneg  14537  zringinvg  14568