Theorem cvgratlem3 8514

Description: Lemma for cvgrati 8517. Restate cvgratlem2 8513 (which was for a real function) in terms of the absolute values of the terms of a complex function F, with the help of an auxiliary function G.

Hypotheses

Ref	Expression
cvgratlem3.1	|- F:NN-->CC
cvgratlem3.2	|- G = {<.y, z>. | (y e. NN /\ z = (abs` (F` y)))}

Assertion

Ref	Expression
cvgratlem3	|- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> ((C e. NN /\ B < C) -> (abs` (F` C)) <_ (((abs` (F` B)) / (A^B)) x. (A^C))))

Distinct variable groups:   x,A   x,y,z,B   x,F,y,z   y,C,z   x,G

Proof of Theorem cvgratlem3

Step	Hyp	Ref	Expression
1		cvgratlem3.2	. . . . . . 7 |- G = {<.y, z>. | (y e. NN /\ z = (abs` (F` y)))}
2		cvgratlem3.1	. . . . . . . . 9 |- F:NN-->CC
3	2	ffvelrni 4788	. . . . . . . 8 |- (y e. NN -> (F` y) e. CC)
4		abscl 8084	. . . . . . . 8 |- ((F` y) e. CC -> (abs` (F` y)) e. RR)
5	3, 4	syl 12	. . . . . . 7 |- (y e. NN -> (abs` (F` y)) e. RR)
6	1, 5	fopab 4800	. . . . . 6 |- G:NN-->RR
7	6	cvgratlem2 8513	. . . . 5 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))))) -> ((C e. NN /\ B < C) -> (G` C) <_ (((G` B) / (A^B)) x. (A^C))))
8	7	imp 377	. . . 4 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))))) /\ (C e. NN /\ B < C)) -> (G` C) <_ (((G` B) / (A^B)) x. (A^C)))
9		fveq2 4681	. . . . . . 7 |- (y = C -> (F` y) = (F` C))
10	9	fveq2d 4685	. . . . . 6 |- (y = C -> (abs` (F` y)) = (abs` (F` C)))
11		fvex 4689	. . . . . 6 |- (abs` (F` C)) e. _V
12	10, 1, 11	fvopab4 4743	. . . . 5 |- (C e. NN -> (G` C) = (abs` (F` C)))
13	12	ad2antrl 442	. . . 4 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))))) /\ (C e. NN /\ B < C)) -> (G` C) = (abs` (F` C)))
14		fveq2 4681	. . . . . . . . . 10 |- (y = B -> (F` y) = (F` B))
15	14	fveq2d 4685	. . . . . . . . 9 |- (y = B -> (abs` (F` y)) = (abs` (F` B)))
16		fvex 4689	. . . . . . . . 9 |- (abs` (F` B)) e. _V
17	15, 1, 16	fvopab4 4743	. . . . . . . 8 |- (B e. NN -> (G` B) = (abs` (F` B)))
18	17	opreq1d 4897	. . . . . . 7 |- (B e. NN -> ((G` B) / (A^B)) = ((abs` (F` B)) / (A^B)))
19	18	opreq1d 4897	. . . . . 6 |- (B e. NN -> (((G` B) / (A^B)) x. (A^C)) = (((abs` (F` B)) / (A^B)) x. (A^C)))
20	19	adantr 425	. . . . 5 |- ((B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x)))) -> (((G` B) / (A^B)) x. (A^C)) = (((abs` (F` B)) / (A^B)) x. (A^C)))
21	20	ad2antlr 441	. . . 4 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))))) /\ (C e. NN /\ B < C)) -> (((G` B) / (A^B)) x. (A^C)) = (((abs` (F` B)) / (A^B)) x. (A^C)))
22	8, 13, 21	3brtr3d 3366	. . 3 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))))) /\ (C e. NN /\ B < C)) -> (abs` (F` C)) <_ (((abs` (F` B)) / (A^B)) x. (A^C)))
23	22	ex 402	. 2 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))))) -> ((C e. NN /\ B < C) -> (abs` (F` C)) <_ (((abs` (F` B)) / (A^B)) x. (A^C))))
24		peano2nn 7118	. . . . . . 7 |- (x e. NN -> (x + 1) e. NN)
25		fveq2 4681	. . . . . . . . 9 |- (y = (x + 1) -> (F` y) = (F` (x + 1)))
26	25	fveq2d 4685	. . . . . . . 8 |- (y = (x + 1) -> (abs` (F` y)) = (abs` (F` (x + 1))))
27		fvex 4689	. . . . . . . 8 |- (abs` (F` (x + 1))) e. _V
28	26, 1, 27	fvopab4 4743	. . . . . . 7 |- ((x + 1) e. NN -> (G` (x + 1)) = (abs` (F` (x + 1))))
29	24, 28	syl 12	. . . . . 6 |- (x e. NN -> (G` (x + 1)) = (abs` (F` (x + 1))))
30		fveq2 4681	. . . . . . . . 9 |- (y = x -> (F` y) = (F` x))
31	30	fveq2d 4685	. . . . . . . 8 |- (y = x -> (abs` (F` y)) = (abs` (F` x)))
32		fvex 4689	. . . . . . . 8 |- (abs` (F` x)) e. _V
33	31, 1, 32	fvopab4 4743	. . . . . . 7 |- (x e. NN -> (G` x) = (abs` (F` x)))
34	33	opreq2d 4898	. . . . . 6 |- (x e. NN -> (A x. (G` x)) = (A x. (abs` (F` x))))
35	29, 34	breq12d 3351	. . . . 5 |- (x e. NN -> ((G` (x + 1)) < (A x. (G` x)) <-> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))
36	35	imbi2d 674	. . . 4 |- (x e. NN -> ((B <_ x -> (G` (x + 1)) < (A x. (G` x))) <-> (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x))))))
37	36	ralbiia 2133	. . 3 |- (A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x))) <-> A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))
38	37	anbi2i 538	. 2 |- ((B e. NN /\ A.x e. NN (B <_ x -> (G` (x + 1)) < (A x. (G` x)))) <-> (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x))))))
39	23, 38	sylan2br 502	1 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> ((C e. NN /\ B < C) -> (abs` (F` C)) <_ (((abs` (F` B)) / (A^B)) x. (A^C))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  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   / cdiv 6447   <_ cle 6448  NNcn 6449   < clt 6653  ^cexp 7811  abscabs 8000

This theorem is referenced by:  cvgratlem5 8516

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-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-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004

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