Theorem gxval 9381

Description: The result of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.)

Hypotheses

Ref	Expression
gxoprval.1	|- X = ran G
gxoprval.2	|- U = (Id` G)
gxoprval.3	|- N = (inv` G)
gxoprval.4	|- P = (^g` G)

Assertion

Ref	Expression
gxval	|- ((G e. Grp /\ A e. X /\ K e. ZZ) -> (APK) = if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))))

Proof of Theorem gxval

Step	Hyp	Ref	Expression
1		gxoprval.1	. . . . 5 |- X = ran G
2		gxoprval.2	. . . . 5 |- U = (Id` G)
3		gxoprval.3	. . . . 5 |- N = (inv` G)
4		gxoprval.4	. . . . 5 |- P = (^g` G)
5	1, 2, 3, 4	gxoprval 9380	. . . 4 |- (G e. Grp -> P = {<.<.x, y>., z>. | ((x e. X /\ y e. ZZ) /\ z = if(y = 0, U, if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy)))))})
6	5	opreqd 4899	. . 3 |- (G e. Grp -> (APK) = (A{<.<.x, y>., z>. | ((x e. X /\ y e. ZZ) /\ z = if(y = 0, U, if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy)))))}K))
7		fvex 4689	. . . . . 6 |- (Id` G) e. _V
8	2, 7	eqeltri 1967	. . . . 5 |- U e. _V
9		fvex 4689	. . . . . 6 |- ((G seq1 (NN X. {A}))` K) e. _V
10		fvex 4689	. . . . . 6 |- (N` ((G seq1 (NN X. {A}))` -uK)) e. _V
11	9, 10	ifex 3031	. . . . 5 |- if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK))) e. _V
12	8, 11	ifex 3031	. . . 4 |- if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))) e. _V
13		sneq 3054	. . . . . . . . . 10 |- (x = A -> {x} = {A})
14		xpeq2 4017	. . . . . . . . . 10 |- ({x} = {A} -> (NN X. {x}) = (NN X. {A}))
15	13, 14	syl 12	. . . . . . . . 9 |- (x = A -> (NN X. {x}) = (NN X. {A}))
16	15	opreq2d 4898	. . . . . . . 8 |- (x = A -> (G seq1 (NN X. {x})) = (G seq1 (NN X. {A})))
17	16	fveq1d 4683	. . . . . . 7 |- (x = A -> ((G seq1 (NN X. {x}))` y) = ((G seq1 (NN X. {A}))` y))
18	17	ifeq1d 2993	. . . . . 6 |- (x = A -> if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy))) = if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {x}))` -uy))))
19	16	fveq1d 4683	. . . . . . . 8 |- (x = A -> ((G seq1 (NN X. {x}))` -uy) = ((G seq1 (NN X. {A}))` -uy))
20	19	fveq2d 4685	. . . . . . 7 |- (x = A -> (N` ((G seq1 (NN X. {x}))` -uy)) = (N` ((G seq1 (NN X. {A}))` -uy)))
21	20	ifeq2d 2994	. . . . . 6 |- (x = A -> if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {x}))` -uy))) = if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy))))
22	18, 21	eqtrd 1925	. . . . 5 |- (x = A -> if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy))) = if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy))))
23	22	ifeq2d 2994	. . . 4 |- (x = A -> if(y = 0, U, if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy)))) = if(y = 0, U, if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy)))))
24		eqeq1 1890	. . . . . 6 |- (y = K -> (y = 0 <-> K = 0))
25	24	ifbid 2996	. . . . 5 |- (y = K -> if(y = 0, U, if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy)))) = if(K = 0, U, if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy)))))
26		breq2 3342	. . . . . . . 8 |- (y = K -> (0 < y <-> 0 < K))
27	26	ifbid 2996	. . . . . . 7 |- (y = K -> if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy))) = if(0 < K, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy))))
28		fveq2 4681	. . . . . . . 8 |- (y = K -> ((G seq1 (NN X. {A}))` y) = ((G seq1 (NN X. {A}))` K))
29	28	ifeq1d 2993	. . . . . . 7 |- (y = K -> if(0 < K, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy))) = if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uy))))
30		negeq 6514	. . . . . . . . . 10 |- (y = K -> -uy = -uK)
31	30	fveq2d 4685	. . . . . . . . 9 |- (y = K -> ((G seq1 (NN X. {A}))` -uy) = ((G seq1 (NN X. {A}))` -uK))
32	31	fveq2d 4685	. . . . . . . 8 |- (y = K -> (N` ((G seq1 (NN X. {A}))` -uy)) = (N` ((G seq1 (NN X. {A}))` -uK)))
33	32	ifeq2d 2994	. . . . . . 7 |- (y = K -> if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uy))) = if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK))))
34	27, 29, 33	3eqtrd 1929	. . . . . 6 |- (y = K -> if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy))) = if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK))))
35	34	ifeq2d 2994	. . . . 5 |- (y = K -> if(K = 0, U, if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy)))) = if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))))
36	25, 35	eqtrd 1925	. . . 4 |- (y = K -> if(y = 0, U, if(0 < y, ((G seq1 (NN X. {A}))` y), (N` ((G seq1 (NN X. {A}))` -uy)))) = if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))))
37		eqid 1884	. . . 4 |- {<.<.x, y>., z>. | ((x e. X /\ y e. ZZ) /\ z = if(y = 0, U, if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy)))))} = {<.<.x, y>., z>. | ((x e. X /\ y e. ZZ) /\ z = if(y = 0, U, if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy)))))}
38	12, 23, 36, 37	oprabval2 4957	. . 3 |- ((A e. X /\ K e. ZZ) -> (A{<.<.x, y>., z>. | ((x e. X /\ y e. ZZ) /\ z = if(y = 0, U, if(0 < y, ((G seq1 (NN X. {x}))` y), (N` ((G seq1 (NN X. {x}))` -uy)))))}K) = if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))))
39	6, 38	sylan9eq 1948	. 2 |- ((G e. Grp /\ (A e. X /\ K e. ZZ)) -> (APK) = if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))))
40	39	3impb 1063	1 |- ((G e. Grp /\ A e. X /\ K e. ZZ) -> (APK) = if(K = 0, U, if(0 < K, ((G seq1 (NN X. {A}))` K), (N` ((G seq1 (NN X. {A}))` -uK)))))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  ifcif 2982  {csn 3044   class class class wbr 3338   X. cxp 3984  ran crn 3987  ` cfv 3998  (class class class)co 4884  {copab2 4885  0cc0 6386  -ucneg 6446  NNcn 6449  ZZcz 6451   < clt 6653   seq1 cseq1 7720  Grpcgr 9311  Idcgi 9312  invcgn 9313  ^gcgx 9315

This theorem is referenced by:  gxpval 9382  gxnval 9383  gx0 9384

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  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  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-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-enr 6318  df-nr 6319  df-0r 6323  df-c 6392  df-r 6396  df-neg 6513  df-z 7345  df-gx 9320

Copyright terms: Public domain