Theorem grpinvfvalNEW 17125

Description: The inverse function of a group.

Hypotheses

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

Assertion

Ref	Expression
grpinvfvalNEW	|- (G e. A -> N = (x e. B |-> (iotay(y e. B /\ (yPx) = U))))

Distinct variable groups:   x,y,B   x,G,y   x,P,y   x,U

Proof of Theorem grpinvfvalNEW

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (G e. A -> G e. _V)
2		fveq2 4681	. . . . . 6 |- (g = G -> (base` g) = (base` G))
3		grpinvfval.1NEW	. . . . . . 7 |- B = (base` G)
4	3	eqcomi 1888	. . . . . 6 |- (base` G) = B
5	2, 4	syl6eq 1944	. . . . 5 |- (g = G -> (base` g) = B)
6	5	eleq2d 1964	. . . . . . 7 |- (g = G -> (y e. (base` g) <-> y e. B))
7		fveq2 4681	. . . . . . . . . 10 |- (g = G -> (+g` g) = (+g` G))
8		grpinvfval.2NEW	. . . . . . . . . 10 |- P = (+g` G)
9	7, 8	syl6eqr 1946	. . . . . . . . 9 |- (g = G -> (+g` g) = P)
10	9	opreqd 4899	. . . . . . . 8 |- (g = G -> (y(+g` g)x) = (yPx))
11		fveq2 4681	. . . . . . . . 9 |- (g = G -> (0g` g) = (0g` G))
12		grpinvfval.3NEW	. . . . . . . . 9 |- U = (0g` G)
13	11, 12	syl6eqr 1946	. . . . . . . 8 |- (g = G -> (0g` g) = U)
14	10, 13	eqeq12d 1899	. . . . . . 7 |- (g = G -> ((y(+g` g)x) = (0g` g) <-> (yPx) = U))
15	6, 14	anbi12d 690	. . . . . 6 |- (g = G -> ((y e. (base` g) /\ (y(+g` g)x) = (0g` g)) <-> (y e. B /\ (yPx) = U)))
16	15	iotabidv 5102	. . . . 5 |- (g = G -> (iotay(y e. (base` g) /\ (y(+g` g)x) = (0g` g))) = (iotay(y e. B /\ (yPx) = U)))
17	5, 16	mpteq12dv 5008	. . . 4 |- (g = G -> (x e. (base` g) |-> (iotay(y e. (base` g) /\ (y(+g` g)x) = (0g` g)))) = (x e. B |-> (iotay(y e. B /\ (yPx) = U))))
18		df-minusg 17091	. . . 4 |- -g = (g e. _V |-> (x e. (base` g) |-> (iotay(y e. (base` g) /\ (y(+g` g)x) = (0g` g)))))
19		fvex 4689	. . . . . 6 |- (base` G) e. _V
20	3, 19	eqeltri 1967	. . . . 5 |- B e. _V
21		mptexg 5012	. . . . 5 |- (B e. _V -> (x e. B |-> (iotay(y e. B /\ (yPx) = U))) e. _V)
22	20, 21	ax-mp 7	. . . 4 |- (x e. B |-> (iotay(y e. B /\ (yPx) = U))) e. _V
23	17, 18, 22	fvmpt 5015	. . 3 |- (G e. _V -> (-g` G) = (x e. B |-> (iotay(y e. B /\ (yPx) = U))))
24		grpinvfval.4NEW	. . 3 |- N = (-g` G)
25	23, 24	syl5eq 1940	. 2 |- (G e. _V -> N = (x e. B |-> (iotay(y e. B /\ (yPx) = U))))
26	1, 25	syl 12	1 |- (G e. A -> N = (x e. B |-> (iotay(y e. B /\ (yPx) = U))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  iotacio 5087  basecbs 16758  +_gcplusg 17080  0_gc0g 17082  -_gcminusg 17083

This theorem is referenced by:  grpinvvalNEW 17126

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-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  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-minusg 17091

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