Theorem grpinvid1 13634

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 6025	. . . 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 13633	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
8	7	3adant3 1043	. . . 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 2266	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( X .+ Y ) = .0. )
11		oveq2 6025	. . . 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 13632	. . . . . . . 8 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
14	13	oveq1d 6032	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
15	14	3adant3 1043	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
16	3, 6	grpinvcl 13630	. . . . . . . . . 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 531	. . . . . . . . 9 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> X e. B )
19		simprr 533	. . . . . . . . 9 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
20	17, 18, 19	3jca 1203	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( N ` X ) e. B /\ X e. B /\ Y e. B ) )
21	3, 4	grpass 13591	. . . . . . . 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 1225	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
24	15, 23	eqtr3d 2266	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
25	3, 4, 5	grplid 13613	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( .0. .+ Y ) = Y )
26	25	3adant2 1042	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = Y )
27	24, 26	eqtr3d 2266	. . . 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 13614	. . . . . 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 1043	. . . 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 2273	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( X .+ Y ) = .0. ) -> ( N ` X ) = Y )
34	10, 33	impbida 600	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 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586

This theorem is referenced by:  grpinvid  13642  grpinvcnv  13650  grpinvadd  13660  subginv  13767  qusinv  13822  ghminv  13836  rngmneg1  13959  ringnegl  14063  lmodindp1  14441  cnfldneg  14586  zringinvg  14617