Theorem grpinvid1 13765

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 6058	. . . 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 13764	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
8	7	3adant3 1044	. . . 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 2267	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( X .+ Y ) = .0. )
11		oveq2 6058	. . . 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 13763	. . . . . . . 8 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
14	13	oveq1d 6065	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
15	14	3adant3 1044	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( .0. .+ Y ) )
16	3, 6	grpinvcl 13761	. . . . . . . . . 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 1204	. . . . . . . 8 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( N ` X ) e. B /\ X e. B /\ Y e. B ) )
21	3, 4	grpass 13722	. . . . . . . 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 1226	. . . . . 6 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( N ` X ) .+ X ) .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
24	15, 23	eqtr3d 2267	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = ( ( N ` X ) .+ ( X .+ Y ) ) )
25	3, 4, 5	grplid 13744	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( .0. .+ Y ) = Y )
26	25	3adant2 1043	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ Y ) = Y )
27	24, 26	eqtr3d 2267	. . . 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 13745	. . . . . 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 1044	. . . 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 2274	. 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 1005   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Grpcgrp 13713   invgcminusg 13714

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717

This theorem is referenced by:  grpinvid  13773  grpinvcnv  13781  grpinvadd  13791  subginv  13898  qusinv  13953  ghminv  13967  rngmneg1  14091  ringnegl  14195  lmodindp1  14576  cnfldneg  14721  zringinvg  14752