Theorem cvgratlem4 7253

Description: Lemma for cvgrat 7255. 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 5931	. . . . . . 7 |- (B e. NN -> B e. RR)
2		leidt 5543	. . . . . . 7 |- (B e. RR -> B <_ B)
3	1, 2	syl 10	. . . . . 6 |- (B e. NN -> B <_ B)
4		breq2 2628	. . . . . . . 8 |- (x = B -> (B <_ x <-> B <_ B))
5		opreq1 3974	. . . . . . . . . . 11 |- (x = B -> (x + 1) = (B + 1))
6	5	fveq2d 3734	. . . . . . . . . 10 |- (x = B -> (F` (x + 1)) = (F` (B + 1)))
7	6	fveq2d 3734	. . . . . . . . 9 |- (x = B -> (abs` (F` (x + 1))) = (abs` (F` (B + 1))))
8		fveq2 3730	. . . . . . . . . . 11 |- (x = B -> (F` x) = (F` B))
9	8	fveq2d 3734	. . . . . . . . . 10 |- (x = B -> (abs` (F` x)) = (abs` (F` B)))
10	9	opreq2d 3982	. . . . . . . . 9 |- (x = B -> (A x. (abs` (F` x))) = (A x. (abs` (F` B))))
11	7, 10	breq12d 2636	. . . . . . . 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 628	. . . . . . 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 1876	. . . . . 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 390	. . . 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 5937	. . . . . . . . 9 |- (B e. NN -> (B + 1) e. NN)
17		cvgrat.1	. . . . . . . . . 10 |- F:NN-->CC
18	17	ffvelrni 3821	. . . . . . . . 9 |- ((B + 1) e. NN -> (F` (B + 1)) e. CC)
19		absge0t 6854	. . . . . . . . 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 390	. . . . . . 7 |- ((A e. RR /\ B e. NN) -> 0 <_ (abs` (F` (B + 1))))
22		0re 5452	. . . . . . . . 9 |- 0 e. RR
23		lelttrt 5535	. . . . . . . . 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 905	. . . . . . . 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 6833	. . . . . . . . . 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 390	. . . . . . . 8 |- ((A e. RR /\ B e. NN) -> (abs` (F` (B + 1))) e. RR)
28		axmulrcl 5286	. . . . . . . . 9 |- ((A e. RR /\ (abs` (F` B)) e. RR) -> (A x. (abs` (F` B))) e. RR)
29	17	ffvelrni 3821	. . . . . . . . . 10 |- (B e. NN -> (F` B) e. CC)
30		absclt 6833	. . . . . . . . . 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 453	. . . . . . . 8 |- ((A e. RR /\ B e. NN) -> (A x. (abs` (F` B))) e. RR)
33	24, 27, 32	sylanc 473	. . . . . . 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 703	. . . . . 6 |- ((A e. RR /\ B e. NN) -> ((abs` (F` (B + 1))) < (A x. (abs` (F` B))) -> 0 < (A x. (abs` (F` B)))))
35		absge0t 6854	. . . . . . . 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 390	. . . . . 6 |- ((A e. RR /\ B e. NN) -> 0 <_ (abs` (F` B)))
38	34, 37	jctild 603	. . . . 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 5829	. . . . . . 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 453	. . . . 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 958   e. wcel 960  A.wral 1648   class class class wbr 2624  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246  1c1 5247   + caddc 5249   x. cmul 5251   <_ cle 5307  NNcn 5308   < clt 5498  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