Theorem grpinvid1 13254

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 5933	. . . 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 13253	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
8	7	3adant3 1019	. . . 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 2231	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( X .+ Y ) = .0. )
11		oveq2 5933	. . . 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 13252	. . . . . . . 8 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
14	13	oveq1d 5940	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
15	14	3adant3 1019	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
16	3, 6	grpinvcl 13250	. . . . . . . . . 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 1179	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( N ` X ) e. B /\ X e. B /\ Y e. B ) )
21	3, 4	grpass 13211	. . . . . . . 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 1201	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
24	15, 23	eqtr3d 2231	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
25	3, 4, 5	grplid 13233	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( .0. .+ Y ) = Y )
26	25	3adant2 1018	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = Y )
27	24, 26	eqtr3d 2231	. . . 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 13234	. . . . . 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 1019	. . . 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 2238	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( X .+ Y ) = .0. ) -> ( N ` X ) = Y )
34	10, 33	impbida 596	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 980   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958   Grpcgrp 13202   invgcminusg 13203

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 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

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 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206

This theorem is referenced by:  grpinvid  13262  grpinvcnv  13270  grpinvadd  13280  subginv  13387  qusinv  13442  ghminv  13456  rngmneg1  13579  ringnegl  13683  lmodindp1  14060  cnfldneg  14205  zringinvg  14236