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Theorem cvgratlem5 8516
Description: Lemma for cvgrati 8517. A complex infinite series F meeting the ratio test criterion converges. We show that the partial sums of F are smaller than the partial sums of a geometric series (which converges by geolimi 8498), so by the comparison test cvgcmp3ce 8451, F also converges.
Hypotheses
Ref Expression
cvgrat.1 |- F:NN-->CC
cvgratlem5.2 |- G = {<.z, w>. | (z e. NN /\ w = (A^z))}
Assertion
Ref Expression
cvgratlem5 |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> E.y( + seq1 F) ~~> y)
Distinct variable groups:   x,y,z,w,A   x,B,y   x,F,y   y,G
Proof of Theorem cvgratlem5
StepHypRef Expression
1 simprl 450 . 2 |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> B e. NN)
2 cvgrat.1 . . . . 5 |- F:NN-->CC
32cvgratlem4 8515 . . . 4 |- ((A e. RR /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> 0 < A)
43adantlr 429 . . 3 |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> 0 < A)
5 0re 6603 . . . . . . . . . . . 12 |- 0 e. RR
6 ltle 6690 . . . . . . . . . . . 12 |- ((0 e. RR /\ A e. RR) -> (0 < A -> 0 <_ A))
75, 6mpan 759 . . . . . . . . . . 11 |- (A e. RR -> (0 < A -> 0 <_ A))
87imp 377 . . . . . . . . . 10 |- ((A e. RR /\ 0 < A) -> 0 <_ A)
98adantlr 429 . . . . . . . . 9 |- (((A e. RR /\ A < 1) /\ 0 < A) -> 0 <_ A)
10 reexpcl 7823 . . . . . . . . . . . . . . . 16 |- ((A e. RR /\ z e. NN0) -> (A^z) e. RR)
11 nnnn0 7315 . . . . . . . . . . . . . . . 16 |- (z e. NN -> z e. NN0)
1210, 11sylan2 500 . . . . . . . . . . . . . . 15 |- ((A e. RR /\ z e. NN) -> (A^z) e. RR)
1312r19.21aiva 2176 . . . . . . . . . . . . . 14 |- (A e. RR -> A.z e. NN (A^z) e. RR)
14 cvgratlem5.2 . . . . . . . . . . . . . . 15 |- G = {<.z, w>. | (z e. NN /\ w = (A^z))}
1514fopab2 4796 . . . . . . . . . . . . . 14 |- (A.z e. NN (A^z) e. RR <-> G:NN-->RR)
1613, 15sylib 215 . . . . . . . . . . . . 13 |- (A e. RR -> G:NN-->RR)
1716adantr 425 . . . . . . . . . . . 12 |- ((A e. RR /\ 0 <_ A) -> G:NN-->RR)
18 expge0 7833 . . . . . . . . . . . . . . . . 17 |- ((A e. RR /\ v e. NN0 /\ 0 <_ A) -> 0 <_ (A^v))
19 nnnn0 7315 . . . . . . . . . . . . . . . . 17 |- (v e. NN -> v e. NN0)
2018, 19syl3an2 1131 . . . . . . . . . . . . . . . 16 |- ((A e. RR /\ v e. NN /\ 0 <_ A) -> 0 <_ (A^v))
21203expa 1067 . . . . . . . . . . . . . . 15 |- (((A e. RR /\ v e. NN) /\ 0 <_ A) -> 0 <_ (A^v))
2221an1rs 547 . . . . . . . . . . . . . 14 |- (((A e. RR /\ 0 <_ A) /\ v e. NN) -> 0 <_ (A^v))
23 opreq2 4890 . . . . . . . . . . . . . . . 16 |- (z = v -> (A^z) = (A^v))
24 oprex 4907 . . . . . . . . . . . . . . . 16 |- (A^v) e. _V
2523, 14, 24fvopab4 4743 . . . . . . . . . . . . . . 15 |- (v e. NN -> (G` v) = (A^v))
2625adantl 424 . . . . . . . . . . . . . 14 |- (((A e. RR /\ 0 <_ A) /\ v e. NN) -> (G` v) = (A^v))
2722, 26breqtrrd 3363 . . . . . . . . . . . . 13 |- (((A e. RR /\ 0 <_ A) /\ v e. NN) -> 0 <_ (G` v))
2827r19.21aiva 2176 . . . . . . . . . . . 12 |- ((A e. RR /\ 0 <_ A) -> A.v e. NN 0 <_ (G` v))
2917, 28jca 310 . . . . . . . . . . 11 |- ((A e. RR /\ 0 <_ A) -> (G:NN-->RR /\ A.v e. NN 0 <_ (G` v)))
3029adantlr 429 . . . . . . . . . 10 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> (G:NN-->RR /\ A.v e. NN 0 <_ (G` v)))
31 recn 6466 . . . . . . . . . . . 12 |- (A e. RR -> A e. CC)
3231ad2antrr 440 . . . . . . . . . . 11 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> A e. CC)
33 absid 8113 . . . . . . . . . . . . 13 |- ((A e. RR /\ 0 <_ A) -> (abs` A) = A)
3433adantlr 429 . . . . . . . . . . . 12 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> (abs` A) = A)
35 simplr 449 . . . . . . . . . . . 12 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> A < 1)
3634, 35eqbrtrd 3357 . . . . . . . . . . 11 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> (abs` A) < 1)
3714geolim1 8501 . . . . . . . . . . 11 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq1 G) ~~> (A / (1 - A)))
3832, 36, 37syl11anc 524 . . . . . . . . . 10 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> ( + seq1 G) ~~> (A / (1 - A)))
3930, 38jca 310 . . . . . . . . 9 |- (((A e. RR /\ A < 1) /\ 0 <_ A) -> ((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))))
409, 39syldan 516 . . . . . . . 8 |- (((A e. RR /\ A < 1) /\ 0 < A) -> ((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))))
4140adantr 425 . . . . . . 7 |- ((((A e. RR /\ A < 1) /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> ((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))))
422ffvelrni 4788 . . . . . . . . . . . . . . . 16 |- (B e. NN -> (F` B) e. CC)
43 abscl 8084 . . . . . . . . . . . . . . . 16 |- ((F` B) e. CC -> (abs` (F` B)) e. RR)
4442, 43syl 12 . . . . . . . . . . . . . . 15 |- (B e. NN -> (abs` (F` B)) e. RR)
45443ad2ant2 898 . . . . . . . . . . . . . 14 |- ((A e. RR /\ B e. NN /\ 0 < A) -> (abs` (F` B)) e. RR)
46 reexpcl 7823 . . . . . . . . . . . . . . . 16 |- ((A e. RR /\ B e. NN0) -> (A^B) e. RR)
47 nnnn0 7315 . . . . . . . . . . . . . . . 16 |- (B e. NN -> B e. NN0)
4846, 47sylan2 500 . . . . . . . . . . . . . . 15 |- ((A e. RR /\ B e. NN) -> (A^B) e. RR)
49483adant3 896 . . . . . . . . . . . . . 14 |- ((A e. RR /\ B e. NN /\ 0 < A) -> (A^B) e. RR)
50 gt0ne0 6800 . . . . . . . . . . . . . . . 16 |- ((A e. RR /\ 0 < A) -> A =/= 0)
51503adant2 895 . . . . . . . . . . . . . . 15 |- ((A e. RR /\ B e. NN /\ 0 < A) -> A =/= 0)
52 expne0 7829 . . . . . . . . . . . . . . . . 17 |- ((A e. CC /\ B e. NN) -> ((A^B) =/= 0 <-> A =/= 0))
5352, 31sylan 497 . . . . . . . . . . . . . . . 16 |- ((A e. RR /\ B e. NN) -> ((A^B) =/= 0 <-> A =/= 0))
54533adant3 896 . . . . . . . . . . . . . . 15 |- ((A e. RR /\ B e. NN /\ 0 < A) -> ((A^B) =/= 0 <-> A =/= 0))
5551, 54mpbird 213 . . . . . . . . . . . . . 14 |- ((A e. RR /\ B e. NN /\ 0 < A) -> (A^B) =/= 0)
56 redivcl 6978 . . . . . . . . . . . . . 14 |- (((abs` (F` B)) e. RR /\ (A^B) e. RR /\ (A^B) =/= 0) -> ((abs` (F` B)) / (A^B)) e. RR)
5745, 49, 55, 56syl111anc 1100 . . . . . . . . . . . . 13 |- ((A e. RR /\ B e. NN /\ 0 < A) -> ((abs` (F` B)) / (A^B)) e. RR)
58 absge0 8105 . . . . . . . . . . . . . . . 16 |- ((F` B) e. CC -> 0 <_ (abs` (F` B)))
5942, 58syl 12 . . . . . . . . . . . . . . 15 |- (B e. NN -> 0 <_ (abs` (F` B)))
60593ad2ant2 898 . . . . . . . . . . . . . 14 |- ((A e. RR /\ B e. NN /\ 0 < A) -> 0 <_ (abs` (F` B)))
61 expgt0 7831 . . . . . . . . . . . . . . 15 |- ((A e. RR /\ B e. NN0 /\ 0 < A) -> 0 < (A^B))
6261, 47syl3an2 1131 . . . . . . . . . . . . . 14 |- ((A e. RR /\ B e. NN /\ 0 < A) -> 0 < (A^B))
63 divge0 7038 . . . . . . . . . . . . . 14 |- ((((abs` (F` B)) e. RR /\ 0 <_ (abs` (F` B))) /\ ((A^B) e. RR /\ 0 < (A^B))) -> 0 <_ ((abs` (F` B)) / (A^B)))
6445, 60, 49, 62, 63syl22anc 1101 . . . . . . . . . . . . 13 |- ((A e. RR /\ B e. NN /\ 0 < A) -> 0 <_ ((abs` (F` B)) / (A^B)))
6557, 64jca 310 . . . . . . . . . . . 12 |- ((A e. RR /\ B e. NN /\ 0 < A) -> (((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))))
66653expa 1067 . . . . . . . . . . 11 |- (((A e. RR /\ B e. NN) /\ 0 < A) -> (((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))))
6766an1rs 547 . . . . . . . . . 10 |- (((A e. RR /\ 0 < A) /\ B e. NN) -> (((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))))
6867adantrr 431 . . . . . . . . 9 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> (((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))))
69 eqid 1884 . . . . . . . . . . . . . . 15 |- {<.y, u>. | (y e. NN /\ u = (abs` (F` y)))} = {<.y, u>. | (y e. NN /\ u = (abs` (F` y)))}
702, 69cvgratlem3 8514 . . . . . . . . . . . . . 14 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> ((v e. NN /\ B < v) -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (A^v))))
7170imp 377 . . . . . . . . . . . . 13 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) /\ (v e. NN /\ B < v)) -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (A^v)))
7225opreq2d 4898 . . . . . . . . . . . . . 14 |- (v e. NN -> (((abs` (F` B)) / (A^B)) x. (G` v)) = (((abs` (F` B)) / (A^B)) x. (A^v)))
7372ad2antrl 442 . . . . . . . . . . . . 13 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) /\ (v e. NN /\ B < v)) -> (((abs` (F` B)) / (A^B)) x. (G` v)) = (((abs` (F` B)) / (A^B)) x. (A^v)))
7471, 73breqtrrd 3363 . . . . . . . . . . . 12 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) /\ (v e. NN /\ B < v)) -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))
7574exp32 408 . . . . . . . . . . 11 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> (v e. NN -> (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))))
7675r19.21aiv 2175 . . . . . . . . . 10 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v))))
7776, 2jctil 316 . . . . . . . . 9 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))))
7868, 77jca 310 . . . . . . . 8 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v))))))
7978adantllr 433 . . . . . . 7 |- ((((A e. RR /\ A < 1) /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v))))))
8041, 79jca 310 . . . . . 6 |- ((((A e. RR /\ A < 1) /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> (((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))) /\ ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))))))
8180exp31 407 . . . . 5 |- ((A e. RR /\ A < 1) -> (0 < A -> ((B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x))))) -> (((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))) /\ ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))))))))
8281com23 36 . . . 4 |- ((A e. RR /\ A < 1) -> ((B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x))))) -> (0 < A -> (((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))) /\ ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))))))))
8382imp 377 . . 3 |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> (0 < A -> (((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))) /\ ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v))))))))
844, 83mpd 29 . 2 |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> (((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))) /\ ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v)))))))
85 oprex 4907 . . 3 |- (A / (1 - A)) e. _V
8685cvgcmp3ce 8451 . 2 |- ((B e. NN /\ (((G:NN-->RR /\ A.v e. NN 0 <_ (G` v)) /\ ( + seq1 G) ~~> (A / (1 - A))) /\ ((((abs` (F` B)) / (A^B)) e. RR /\ 0 <_ ((abs` (F` B)) / (A^B))) /\ (F:NN-->CC /\ A.v e. NN (B < v -> (abs` (F` v)) <_ (((abs` (F` B)) / (A^B)) x. (G` v))))))) -> E.y( + seq1 F) ~~> y)
871, 84, 86syl11anc 524 1 |- (((A e. RR /\ A < 1) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> E.y( + seq1 F) ~~> y)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105   class class class wbr 3338  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653   seq1 cseq1 7720  ^cexp 7811  abscabs 8000   ~~> cli 8234
This theorem is referenced by:  cvgrati 8517
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-nel 2020  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  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-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240
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