Theorem grpinvid2 13586

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 6008	. . . 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 13583	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
8	7	3adant3 1041	. . . 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 2264	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( Y .+ X ) = .0. )
11	3, 6	grpinvcl 13581	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
12	3, 4, 5	grplid 13564	. . . . . . 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 1041	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ ( N ` X ) ) = ( N ` X ) )
15	14	eqcomd 2235	. . . 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 6008	. . . 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 1201	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( Y e. B /\ X e. B /\ ( N ` X ) e. B ) )
23	3, 4	grpass 13542	. . . . . . 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 1223	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( Y .+ ( X .+ ( N ` X ) ) ) )
26	3, 4, 5, 6	grprinv 13584	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
27	26	oveq2d 6017	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( Y .+ ( X .+ ( N ` X ) ) ) = ( Y .+ .0. ) )
28	27	3adant3 1041	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .+ ( X .+ ( N ` X ) ) ) = ( Y .+ .0. ) )
29	3, 4, 5	grprid 13565	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( Y .+ .0. ) = Y )
30	29	3adant2 1040	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .+ .0. ) = Y )
31	25, 28, 30	3eqtrd 2266	. . . 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 2268	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( Y .+ X ) = .0. ) -> ( N ` X ) = Y )
34	10, 33	impbida 598	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 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:  grpinvcnv  13601  grpsubeq0  13619  prdsinvgd  13643  rngmneg2  13911  ringnegr  14015  psrneg  14651