Theorem cvgratlem4 7196

Description: Lemma for cvgrat 7198. 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 5885	. . . . . . 7 |- (B e. NN -> B e. RR)
2		leidt 5512	. . . . . . 7 |- (B e. RR -> B <_ B)
3	1, 2	syl 10	. . . . . 6 |- (B e. NN -> B <_ B)
4		breq2 2618	. . . . . . . 8 |- (x = B -> (B <_ x <-> B <_ B))
5		opreq1 3959	. . . . . . . . . . 11 |- (x = B -> (x + 1) = (B + 1))
6	5	fveq2d 3719	. . . . . . . . . 10 |- (x = B -> (F` (x + 1)) = (F` (B + 1)))
7	6	fveq2d 3719	. . . . . . . . 9 |- (x = B -> (abs` (F` (x + 1))) = (abs` (F` (B + 1))))
8		fveq2 3715	. . . . . . . . . . 11 |- (x = B -> (F` x) = (F` B))
9	8	fveq2d 3719	. . . . . . . . . 10 |- (x = B -> (abs` (F` x)) = (abs` (F` B)))
10	9	opreq2d 3967	. . . . . . . . 9 |- (x = B -> (A x. (abs` (F` x))) = (A x. (abs` (F` B))))
11	7, 10	breq12d 2626	. . . . . . . 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 1869	. . . . . 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 5891	. . . . . . . . 9 |- (B e. NN -> (B + 1) e. NN)
17		cvgrat.1	. . . . . . . . . 10 |- F:NN-->CC
18	17	ffvelrni 3806	. . . . . . . . 9 |- ((B + 1) e. NN -> (F` (B + 1)) e. CC)
19		absge0t 6797	. . . . . . . . 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 5420	. . . . . . . . 9 |- 0 e. RR
23		lelttrt 5504	. . . . . . . . 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 901	. . . . . . . 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 6776	. . . . . . . . . 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 5254	. . . . . . . . 9 |- ((A e. RR /\ (abs` (F` B)) e. RR) -> (A x. (abs` (F` B))) e. RR)
29	17	ffvelrni 3806	. . . . . . . . . 10 |- (B e. NN -> (F` B) e. CC)
30		absclt 6776	. . . . . . . . . 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 6797	. . . . . . . 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 5791	. . . . . . 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 954   e. wcel 956  A.wral 1642   class class class wbr 2614  -->wf 3173  ` cfv 3177  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214  1c1 5215   + caddc 5217   x. cmul 5219   <_ cle 5275  NNcn 5276   < clt 5466  abscabs 6689

This theorem is referenced by:  cvgratlem5 7197

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp