Theorem grpinvid1NEW 17131

Description: The inverse of a group element expressed in terms of the identity element.

Hypotheses

Ref	Expression
grpinv.1NEW	|- B = (base` G)
grpinv.2NEW	|- P = (+g` G)
grpinv.3NEW	|- U = (0g` G)
grpinv.4NEW	|- N = (-g` G)

Assertion

Ref	Expression
grpinvid1NEW	|- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> ((N` X) = Y <-> (XPY) = U))

Proof of Theorem grpinvid1NEW

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . 4 |- ((N` X) = Y -> (XP(N` X)) = (XPY))
2	1	adantl 424	. . 3 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (N` X) = Y) -> (XP(N` X)) = (XPY))
3		grpinv.1NEW	. . . . . 6 |- B = (base` G)
4		grpinv.2NEW	. . . . . 6 |- P = (+g` G)
5		grpinv.3NEW	. . . . . 6 |- U = (0g` G)
6		grpinv.4NEW	. . . . . 6 |- N = (-g` G)
7	3, 4, 5, 6	grprinvNEW 17130	. . . . 5 |- ((G e. GrpNEW /\ X e. B) -> (XP(N` X)) = U)
8	7	3adant3 896	. . . 4 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> (XP(N` X)) = U)
9	8	adantr 425	. . 3 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (N` X) = Y) -> (XP(N` X)) = U)
10	2, 9	eqtr3d 1927	. 2 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (N` X) = Y) -> (XPY) = U)
11		opreq2 4890	. . . 4 |- ((XPY) = U -> ((N` X)P(XPY)) = ((N` X)PU))
12	11	adantl 424	. . 3 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (XPY) = U) -> ((N` X)P(XPY)) = ((N` X)PU))
13	3, 4, 5, 6	grplinvNEW 17129	. . . . . . 7 |- ((G e. GrpNEW /\ X e. B) -> ((N` X)PX) = U)
14	13	opreq1d 4897	. . . . . 6 |- ((G e. GrpNEW /\ X e. B) -> (((N` X)PX)PY) = (UPY))
15	14	3adant3 896	. . . . 5 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> (((N` X)PX)PY) = (UPY))
16	3, 4, 6	grpinvclNEW 17127	. . . . . . . . 9 |- ((G e. GrpNEW /\ X e. B) -> (N` X) e. B)
17	16	adantrr 431	. . . . . . . 8 |- ((G e. GrpNEW /\ (X e. B /\ Y e. B)) -> (N` X) e. B)
18		simprl 450	. . . . . . . 8 |- ((G e. GrpNEW /\ (X e. B /\ Y e. B)) -> X e. B)
19		simprr 451	. . . . . . . 8 |- ((G e. GrpNEW /\ (X e. B /\ Y e. B)) -> Y e. B)
20	17, 18, 19	3jca 1050	. . . . . . 7 |- ((G e. GrpNEW /\ (X e. B /\ Y e. B)) -> ((N` X) e. B /\ X e. B /\ Y e. B))
21	3, 4	grpassNEW 17107	. . . . . . 7 |- ((G e. GrpNEW /\ ((N` X) e. B /\ X e. B /\ Y e. B)) -> (((N` X)PX)PY) = ((N` X)P(XPY)))
22	20, 21	syldan 516	. . . . . 6 |- ((G e. GrpNEW /\ (X e. B /\ Y e. B)) -> (((N` X)PX)PY) = ((N` X)P(XPY)))
23	22	3impb 1063	. . . . 5 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> (((N` X)PX)PY) = ((N` X)P(XPY)))
24	3, 4, 5	grplidNEW 17120	. . . . . 6 |- ((G e. GrpNEW /\ Y e. B) -> (UPY) = Y)
25	24	3adant2 895	. . . . 5 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> (UPY) = Y)
26	15, 23, 25	3eqtr3d 1934	. . . 4 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> ((N` X)P(XPY)) = Y)
27	26	adantr 425	. . 3 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (XPY) = U) -> ((N` X)P(XPY)) = Y)
28	3, 4, 5	grpridNEW 17121	. . . . . 6 |- ((G e. GrpNEW /\ (N` X) e. B) -> ((N` X)PU) = (N` X))
29	16, 28	syldan 516	. . . . 5 |- ((G e. GrpNEW /\ X e. B) -> ((N` X)PU) = (N` X))
30	29	3adant3 896	. . . 4 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> ((N` X)PU) = (N` X))
31	30	adantr 425	. . 3 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (XPY) = U) -> ((N` X)PU) = (N` X))
32	12, 27, 31	3eqtr3rd 1936	. 2 |- (((G e. GrpNEW /\ X e. B /\ Y e. B) /\ (XPY) = U) -> (N` X) = Y)
33	10, 32	impbida 577	1 |- ((G e. GrpNEW /\ X e. B /\ Y e. B) -> ((N` X) = Y <-> (XPY) = U))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081  0_gc0g 17082  -_gcminusg 17083

This theorem is referenced by:  grpinvidNEW 17133

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-mpt 5006  df-iota 5089  df-struct 16708  df-grpNEW 17089  df-0g 17090  df-minusg 17091

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