Theorem grpinvNEW 17128

Description: The properties of a group element's inverse.

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
grpinvNEW	|- ((G e. GrpNEW /\ X e. B) -> (((N` X)PX) = U /\ (XP(N` X)) = U))

Proof of Theorem grpinvNEW

Step	Hyp	Ref	Expression
1		grpinv.1NEW	. . . . . 6 |- B = (base` G)
2		grpinv.2NEW	. . . . . 6 |- P = (+g` G)
3		grpinv.3NEW	. . . . . 6 |- U = (0g` G)
4	1, 2, 3	grpidinv2NEW 17119	. . . . 5 |- ((G e. GrpNEW /\ X e. B) -> (((UPX) = X /\ (XPU) = X) /\ E.y e. B ((yPX) = U /\ (XPy) = U)))
5	4	simprd 352	. . . 4 |- ((G e. GrpNEW /\ X e. B) -> E.y e. B ((yPX) = U /\ (XPy) = U))
6	1, 2, 3	grpinveuNEW 17123	. . . 4 |- ((G e. GrpNEW /\ X e. B) -> E!y e. B (yPX) = U)
7		ssid 2634	. . . . 5 |- B C_ B
8		simpl 346	. . . . . . 7 |- (((yPX) = U /\ (XPy) = U) -> (yPX) = U)
9	8	a1i 8	. . . . . 6 |- (y e. B -> (((yPX) = U /\ (XPy) = U) -> (yPX) = U))
10	9	rgen 2159	. . . . 5 |- A.y e. B (((yPX) = U /\ (XPy) = U) -> (yPX) = U)
11		reiotass2 5111	. . . . 5 |- (((B C_ B /\ A.y e. B (((yPX) = U /\ (XPy) = U) -> (yPX) = U)) /\ (E.y e. B ((yPX) = U /\ (XPy) = U) /\ E!y e. B (yPX) = U)) -> (iotay(y e. B /\ ((yPX) = U /\ (XPy) = U))) = (iotay(y e. B /\ (yPX) = U)))
12	7, 10, 11	mpanl12 773	. . . 4 |- ((E.y e. B ((yPX) = U /\ (XPy) = U) /\ E!y e. B (yPX) = U) -> (iotay(y e. B /\ ((yPX) = U /\ (XPy) = U))) = (iotay(y e. B /\ (yPX) = U)))
13	5, 6, 12	syl11anc 524	. . 3 |- ((G e. GrpNEW /\ X e. B) -> (iotay(y e. B /\ ((yPX) = U /\ (XPy) = U))) = (iotay(y e. B /\ (yPX) = U)))
14		grpinv.4NEW	. . . 4 |- N = (-g` G)
15	1, 2, 3, 14	grpinvvalNEW 17126	. . 3 |- ((G e. GrpNEW /\ X e. B) -> (N` X) = (iotay(y e. B /\ (yPX) = U)))
16	13, 15	eqtr4d 1928	. 2 |- ((G e. GrpNEW /\ X e. B) -> (iotay(y e. B /\ ((yPX) = U /\ (XPy) = U))) = (N` X))
17	1, 2, 14	grpinvclNEW 17127	. . 3 |- ((G e. GrpNEW /\ X e. B) -> (N` X) e. B)
18		reuss2 2870	. . . . 5 |- (((B C_ B /\ A.y e. B (((yPX) = U /\ (XPy) = U) -> (yPX) = U)) /\ (E.y e. B ((yPX) = U /\ (XPy) = U) /\ E!y e. B (yPX) = U)) -> E!y e. B ((yPX) = U /\ (XPy) = U))
19	7, 10, 18	mpanl12 773	. . . 4 |- ((E.y e. B ((yPX) = U /\ (XPy) = U) /\ E!y e. B (yPX) = U) -> E!y e. B ((yPX) = U /\ (XPy) = U))
20	5, 6, 19	syl11anc 524	. . 3 |- ((G e. GrpNEW /\ X e. B) -> E!y e. B ((yPX) = U /\ (XPy) = U))
21		opreq1 4889	. . . . . 6 |- (y = (N` X) -> (yPX) = ((N` X)PX))
22	21	eqeq1d 1892	. . . . 5 |- (y = (N` X) -> ((yPX) = U <-> ((N` X)PX) = U))
23		opreq2 4890	. . . . . 6 |- (y = (N` X) -> (XPy) = (XP(N` X)))
24	23	eqeq1d 1892	. . . . 5 |- (y = (N` X) -> ((XPy) = U <-> (XP(N` X)) = U))
25	22, 24	anbi12d 690	. . . 4 |- (y = (N` X) -> (((yPX) = U /\ (XPy) = U) <-> (((N` X)PX) = U /\ (XP(N` X)) = U)))
26	25	reiota2 5110	. . 3 |- (((N` X) e. B /\ E!y e. B ((yPX) = U /\ (XPy) = U)) -> ((((N` X)PX) = U /\ (XP(N` X)) = U) <-> (iotay(y e. B /\ ((yPX) = U /\ (XPy) = U))) = (N` X)))
27	17, 20, 26	syl11anc 524	. 2 |- ((G e. GrpNEW /\ X e. B) -> ((((N` X)PX) = U /\ (XP(N` X)) = U) <-> (iotay(y e. B /\ ((yPX) = U /\ (XPy) = U))) = (N` X)))
28	16, 27	mpbird 213	1 |- ((G e. GrpNEW /\ X e. B) -> (((N` X)PX) = U /\ (XP(N` X)) = U))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107   C_ wss 2593  ` cfv 3998  (class class class)co 4884  iotacio 5087  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081  0_gc0g 17082  -_gcminusg 17083

This theorem is referenced by:  grplinvNEW 17129  grprinvNEW 17130

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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