Theorem grpinvid2 13500

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 5974	. . . 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 13497	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
8	7	3adant3 1020	. . . 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 2242	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( N ` X ) = Y ) -> ( Y .+ X ) = .0. )
11	3, 6	grpinvcl 13495	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
12	3, 4, 5	grplid 13478	. . . . . . 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 1020	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( .0. .+ ( N ` X ) ) = ( N ` X ) )
15	14	eqcomd 2213	. . . 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 5974	. . . 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 1180	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( Y e. B /\ X e. B /\ ( N ` X ) e. B ) )
23	3, 4	grpass 13456	. . . . . . 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 1202	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( Y .+ X ) .+ ( N ` X ) ) = ( Y .+ ( X .+ ( N ` X ) ) ) )
26	3, 4, 5, 6	grprinv 13498	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
27	26	oveq2d 5983	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( Y .+ ( X .+ ( N ` X ) ) ) = ( Y .+ .0. ) )
28	27	3adant3 1020	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .+ ( X .+ ( N ` X ) ) ) = ( Y .+ .0. ) )
29	3, 4, 5	grprid 13479	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( Y .+ .0. ) = Y )
30	29	3adant2 1019	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .+ .0. ) = Y )
31	25, 28, 30	3eqtrd 2244	. . . 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 2246	. 2 |- ( ( ( G e. Grp /\ X e. B /\ Y e. B ) /\ ( Y .+ X ) = .0. ) -> ( N ` X ) = Y )
34	10, 33	impbida 596	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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447   invgcminusg 13448

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451

This theorem is referenced by:  grpinvcnv  13515  grpsubeq0  13533  prdsinvgd  13557  rngmneg2  13825  ringnegr  13929  psrneg  14564