Theorem cvgratlem4 7205

Description: Lemma for cvgrat 7207. The ratio of successive terms meeting the ratio test criterion is positive.

Hypothesis

Ref	Expression
cvgrat.1	|- F:NN-->CC

Assertion

Ref	Expression
cvgratlem4	|- ((A e. RR /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> 0 < A)

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem cvgratlem4

Step	Hyp	Ref	Expression
1		nnret 5887	. . . . . . 7 |- (B e. NN -> B e. RR)
2		leidt 5514	. . . . . . 7 |- (B e. RR -> B <_ B)
3	1, 2	syl 10	. . . . . 6 |- (B e. NN -> B <_ B)
4		breq2 2619	. . . . . . . 8 |- (x = B -> (B <_ x <-> B <_ B))
5		opreq1 3963	. . . . . . . . . . 11 |- (x = B -> (x + 1) = (B + 1))
6	5	fveq2d 3723	. . . . . . . . . 10 |- (x = B -> (F` (x + 1)) = (F` (B + 1)))
7	6	fveq2d 3723	. . . . . . . . 9 |- (x = B -> (abs` (F` (x + 1))) = (abs` (F` (B + 1))))
8		fveq2 3719	. . . . . . . . . . 11 |- (x = B -> (F` x) = (F` B))
9	8	fveq2d 3723	. . . . . . . . . 10 |- (x = B -> (abs` (F` x)) = (abs` (F` B)))
10	9	opreq2d 3971	. . . . . . . . 9 |- (x = B -> (A x. (abs` (F` x))) = (A x. (abs` (F` B))))
11	7, 10	breq12d 2627	. . . . . . . 8 |- (x = B -> ((abs` (F` (x + 1))) < (A x. (abs` (F` x))) <-> (abs` (F` (B + 1))) < (A x. (abs` (F` B)))))
12	4, 11	imbi12d 625	. . . . . . 7 |- (x = B -> ((B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))) <-> (B <_ B -> (abs` (F` (B + 1))) < (A x. (abs` (F` B))))))
13	12	rcla4v 1870	. . . . . 6 |- (B e. NN -> (A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))) -> (B <_ B -> (abs` (F` (B + 1))) < (A x. (abs` (F` B))))))
14	3, 13	mpid 47	. . . . 5 |- (B e. NN -> (A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))) -> (abs` (F` (B + 1))) < (A x. (abs` (F` B)))))
15	14	adantl 388	. . . 4 |- ((A e. RR /\ B e. NN) -> (A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))) -> (abs` (F` (B + 1))) < (A x. (abs` (F` B)))))
16		peano2nn 5893	. . . . . . . . 9 |- (B e. NN -> (B + 1) e. NN)
17		cvgrat.1	. . . . . . . . . 10 |- F:NN-->CC
18	17	ffvelrni 3810	. . . . . . . . 9 |- ((B + 1) e. NN -> (F` (B + 1)) e. CC)
19		absge0t 6804	. . . . . . . . 9 |- ((F` (B + 1)) e. CC -> 0 <_ (abs` (F` (B + 1))))
20	16, 18, 19	3syl 20	. . . . . . . 8 |- (B e. NN -> 0 <_ (abs` (F` (B + 1))))
21	20	adantl 388	. . . . . . 7 |- ((A e. RR /\ B e. NN) -> 0 <_ (abs` (F` (B + 1))))
22		0re 5423	. . . . . . . . 9 |- 0 e. RR
23		lelttrt 5506	. . . . . . . . 9 |- ((0 e. RR /\ (abs` (F` (B + 1))) e. RR /\ (A x. (abs` (F` B))) e. RR) -> ((0 <_ (abs` (F` (B + 1))) /\ (abs` (F` (B + 1))) < (A x. (abs` (F` B)))) -> 0 < (A x. (abs` (F` B)))))
24	22, 23	mp3an1 902	. . . . . . . 8 |- (((abs` (F` (B + 1))) e. RR /\ (A x. (abs` (F` B))) e. RR) -> ((0 <_ (abs` (F` (B + 1))) /\ (abs` (F` (B + 1))) < (A x. (abs` (F` B)))) -> 0 < (A x. (abs` (F` B)))))
25		absclt 6783	. . . . . . . . . 10 |- ((F` (B + 1)) e. CC -> (abs` (F` (B + 1))) e. RR)
26	16, 18, 25	3syl 20	. . . . . . . . 9 |- (B e. NN -> (abs` (F` (B + 1))) e. RR)
27	26	adantl 388	. . . . . . . 8 |- ((A e. RR /\ B e. NN) -> (abs` (F` (B + 1))) e. RR)
28		axmulrcl 5257	. . . . . . . . 9 |- ((A e. RR /\ (abs` (F` B)) e. RR) -> (A x. (abs` (F` B))) e. RR)
29	17	ffvelrni 3810	. . . . . . . . . 10 |- (B e. NN -> (F` B) e. CC)
30		absclt 6783	. . . . . . . . . 10 |- ((F` B) e. CC -> (abs` (F` B)) e. RR)
31	29, 30	syl 10	. . . . . . . . 9 |- (B e. NN -> (abs` (F` B)) e. RR)
32	28, 31	sylan2 451	. . . . . . . 8 |- ((A e. RR /\ B e. NN) -> (A x. (abs` (F` B))) e. RR)
33	24, 27, 32	sylanc 471	. . . . . . 7 |- ((A e. RR /\ B e. NN) -> ((0 <_ (abs` (F` (B + 1))) /\ (abs` (F` (B + 1))) < (A x. (abs` (F` B)))) -> 0 < (A x. (abs` (F` B)))))
34	21, 33	mpand 700	. . . . . 6 |- ((A e. RR /\ B e. NN) -> ((abs` (F` (B + 1))) < (A x. (abs` (F` B))) -> 0 < (A x. (abs` (F` B)))))
35		absge0t 6804	. . . . . . . 8 |- ((F` B) e. CC -> 0 <_ (abs` (F` B)))
36	29, 35	syl 10	. . . . . . 7 |- (B e. NN -> 0 <_ (abs` (F` B)))
37	36	adantl 388	. . . . . 6 |- ((A e. RR /\ B e. NN) -> 0 <_ (abs` (F` B)))
38	34, 37	jctild 600	. . . . 5 |- ((A e. RR /\ B e. NN) -> ((abs` (F` (B + 1))) < (A x. (abs` (F` B))) -> (0 <_ (abs` (F` B)) /\ 0 < (A x. (abs` (F` B))))))
39		prodgt02t 5793	. . . . . . 7 |- (((A e. RR /\ (abs` (F` B)) e. RR) /\ (0 <_ (abs` (F` B)) /\ 0 < (A x. (abs` (F` B))))) -> 0 < A)
40	39	ex 373	. . . . . 6 |- ((A e. RR /\ (abs` (F` B)) e. RR) -> ((0 <_ (abs` (F` B)) /\ 0 < (A x. (abs` (F` B)))) -> 0 < A))
41	40, 31	sylan2 451	. . . . 5 |- ((A e. RR /\ B e. NN) -> ((0 <_ (abs` (F` B)) /\ 0 < (A x. (abs` (F` B)))) -> 0 < A))
42	38, 41	syld 27	. . . 4 |- ((A e. RR /\ B e. NN) -> ((abs` (F` (B + 1))) < (A x. (abs` (F` B))) -> 0 < A))
43	15, 42	syld 27	. . 3 |- ((A e. RR /\ B e. NN) -> (A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))) -> 0 < A))
44	43	ex 373	. 2 |- (A e. RR -> (B e. NN -> (A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))) -> 0 < A)))
45	44	imp32 363	1 |- ((A e. RR /\ (B e. NN /\ A.x e. NN (B <_ x -> (abs` (F` (x + 1))) < (A x. (abs` (F` x)))))) -> 0 < A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643   class class class wbr 2615  -->wf 3174  ` cfv 3178  (class class class)co 3958  CCcc 5215  RRcr 5216  0cc0 5217  1c1 5218   + caddc 5220   x. cmul 5222   <_ cle 5278  NNcn 5279   < clt 5469  abscabs 6696

This theorem is referenced by:  cvgratlem5 7206

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-sup 4557  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp