Theorem cvgratlem3 7252

Description: Lemma for cvgrat 7255. Restate cvgratlem2 7251 (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 3821	. . . . . . . 8 |- (y e. NN -> (F` y) e. CC)
4		absclt 6833	. . . . . . . 8 |- ((F` y) e. CC -> (abs` (F` y)) e. RR)
5	3, 4	syl 10	. . . . . . 7 |- (y e. NN -> (abs` (F` y)) e. RR)
6	1, 5	fopab 3833	. . . . . 6 |- G:NN-->RR
7	6	cvgratlem2 7251	. . . . 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 350	. . . 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 3730	. . . . . . 7 |- (y = C -> (F` y) = (F` C))
10	9	fveq2d 3734	. . . . . 6 |- (y = C -> (abs` (F` y)) = (abs` (F` C)))
11		fvex 3738	. . . . . 6 |- (abs` (F` C)) e. V
12	10, 1, 11	fvopab4 3786	. . . . 5 |- (C e. NN -> (G` C) = (abs` (F` C)))
13	12	ad2antrl 408	. . . 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 3730	. . . . . . . . . 10 |- (y = B -> (F` y) = (F` B))
15	14	fveq2d 3734	. . . . . . . . 9 |- (y = B -> (abs` (F` y)) = (abs` (F` B)))
16		fvex 3738	. . . . . . . . 9 |- (abs` (F` B)) e. V
17	15, 1, 16	fvopab4 3786	. . . . . . . 8 |- (B e. NN -> (G` B) = (abs` (F` B)))
18	17	opreq1d 3981	. . . . . . 7 |- (B e. NN -> ((G` B) / (A^B)) = ((abs` (F` B)) / (A^B)))
19	18	opreq1d 3981	. . . . . 6 |- (B e. NN -> (((G` B) / (A^B)) x. (A^C)) = (((abs` (F` B)) / (A^B)) x. (A^C)))
20	19	adantr 391	. . . . 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 407	. . . 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 2649	. . 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 373	. 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 5937	. . . . . . 7 |- (x e. NN -> (x + 1) e. NN)
25		fveq2 3730	. . . . . . . . 9 |- (y = (x + 1) -> (F` y) = (F` (x + 1)))
26	25	fveq2d 3734	. . . . . . . 8 |- (y = (x + 1) -> (abs` (F` y)) = (abs` (F` (x + 1))))
27		fvex 3738	. . . . . . . 8 |- (abs` (F` (x + 1))) e. V
28	26, 1, 27	fvopab4 3786	. . . . . . 7 |- ((x + 1) e. NN -> (G` (x + 1)) = (abs` (F` (x + 1))))
29	24, 28	syl 10	. . . . . 6 |- (x e. NN -> (G` (x + 1)) = (abs` (F` (x + 1))))
30		fveq2 3730	. . . . . . . . 9 |- (y = x -> (F` y) = (F` x))
31	30	fveq2d 3734	. . . . . . . 8 |- (y = x -> (abs` (F` y)) = (abs` (F` x)))
32		fvex 3738	. . . . . . . 8 |- (abs` (F` x)) e. V
33	31, 1, 32	fvopab4 3786	. . . . . . 7 |- (x e. NN -> (G` x) = (abs` (F` x)))
34	33	opreq2d 3982	. . . . . 6 |- (x e. NN -> (A x. (G` x)) = (A x. (abs` (F` x))))
35	29, 34	breq12d 2636	. . . . 5 |- (x e. NN -> ((G` (x + 1)) < (A x. (G` x)) <-> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))
36	35	imbi2d 614	. . . 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 1676	. . 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 482	. 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 455	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 223   = wceq 958   e. wcel 960  A.wral 1648   class class class wbr 2624  {copab 2671  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246  1c1 5247   + caddc 5249   x. cmul 5251   / cdiv 5306   <_ cle 5307  NNcn 5308   < clt 5498  ^cexp 6569  abscabs 6751

This theorem is referenced by:  cvgratlem5 7254

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500