Theorem cvgratlem2 8513

Description: Lemma for cvgrati 8517. Using expsubOLD 7842, restate cvgratlem1 8512 with an absolute index C instead of just an offset from the starting index B.

Hypothesis

Ref	Expression
cvgratlem1.1	|- F:NN-->RR

Assertion

Ref	Expression
cvgratlem2	|- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> ((C e. NN /\ B < C) -> (F` C) <_ (((F` B) / (A^B)) x. (A^C))))

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

Proof of Theorem cvgratlem2

Step	Hyp	Ref	Expression
1		nnsub 7141	. . . . . . . . . 10 |- ((B e. NN /\ C e. NN) -> (B < C <-> (C - B) e. NN))
2		cvgratlem1.1	. . . . . . . . . . . . . . . . 17 |- F:NN-->RR
3	2	cvgratlem1 8512	. . . . . . . . . . . . . . . 16 |- (((C - B) e. NN /\ ((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))))) -> (F` (B + (C - B))) <_ ((A^(C - B)) x. (F` B)))
4	3	exp45 417	. . . . . . . . . . . . . . 15 |- ((C - B) e. NN -> ((A e. RR /\ 0 < A) -> (B e. NN -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` (B + (C - B))) <_ ((A^(C - B)) x. (F` B))))))
5	4	com3r 39	. . . . . . . . . . . . . 14 |- (B e. NN -> ((C - B) e. NN -> ((A e. RR /\ 0 < A) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` (B + (C - B))) <_ ((A^(C - B)) x. (F` B))))))
6	5	imp4d 394	. . . . . . . . . . . . 13 |- (B e. NN -> (((C - B) e. NN /\ ((A e. RR /\ 0 < A) /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + (C - B))) <_ ((A^(C - B)) x. (F` B))))
7	6	adantr 425	. . . . . . . . . . . 12 |- ((B e. NN /\ C e. NN) -> (((C - B) e. NN /\ ((A e. RR /\ 0 < A) /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + (C - B))) <_ ((A^(C - B)) x. (F` B))))
8		pncan3 6534	. . . . . . . . . . . . . . 15 |- ((B e. CC /\ C e. CC) -> (B + (C - B)) = C)
9		nncn 7113	. . . . . . . . . . . . . . 15 |- (B e. NN -> B e. CC)
10		nncn 7113	. . . . . . . . . . . . . . 15 |- (C e. NN -> C e. CC)
11	8, 9, 10	syl2an 503	. . . . . . . . . . . . . 14 |- ((B e. NN /\ C e. NN) -> (B + (C - B)) = C)
12	11	fveq2d 4685	. . . . . . . . . . . . 13 |- ((B e. NN /\ C e. NN) -> (F` (B + (C - B))) = (F` C))
13	12	breq1d 3348	. . . . . . . . . . . 12 |- ((B e. NN /\ C e. NN) -> ((F` (B + (C - B))) <_ ((A^(C - B)) x. (F` B)) <-> (F` C) <_ ((A^(C - B)) x. (F` B))))
14	7, 13	sylibd 219	. . . . . . . . . . 11 |- ((B e. NN /\ C e. NN) -> (((C - B) e. NN /\ ((A e. RR /\ 0 < A) /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` C) <_ ((A^(C - B)) x. (F` B))))
15	14	exp3a 405	. . . . . . . . . 10 |- ((B e. NN /\ C e. NN) -> ((C - B) e. NN -> (((A e. RR /\ 0 < A) /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))) -> (F` C) <_ ((A^(C - B)) x. (F` B)))))
16	1, 15	sylbid 220	. . . . . . . . 9 |- ((B e. NN /\ C e. NN) -> (B < C -> (((A e. RR /\ 0 < A) /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))) -> (F` C) <_ ((A^(C - B)) x. (F` B)))))
17	16	expimpd 404	. . . . . . . 8 |- (B e. NN -> ((C e. NN /\ B < C) -> (((A e. RR /\ 0 < A) /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))) -> (F` C) <_ ((A^(C - B)) x. (F` B)))))
18	17	exp4a 409	. . . . . . 7 |- (B e. NN -> ((C e. NN /\ B < C) -> ((A e. RR /\ 0 < A) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` C) <_ ((A^(C - B)) x. (F` B))))))
19	18	com12 14	. . . . . 6 |- ((C e. NN /\ B < C) -> (B e. NN -> ((A e. RR /\ 0 < A) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` C) <_ ((A^(C - B)) x. (F` B))))))
20	19	com4l 43	. . . . 5 |- (B e. NN -> ((A e. RR /\ 0 < A) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> ((C e. NN /\ B < C) -> (F` C) <_ ((A^(C - B)) x. (F` B))))))
21	20	com12 14	. . . 4 |- ((A e. RR /\ 0 < A) -> (B e. NN -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> ((C e. NN /\ B < C) -> (F` C) <_ ((A^(C - B)) x. (F` B))))))
22	21	imp42 396	. . 3 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) /\ (C e. NN /\ B < C)) -> (F` C) <_ ((A^(C - B)) x. (F` B)))
23		recn 6466	. . . . . . . . . . . 12 |- (A e. RR -> A e. CC)
24		id 73	. . . . . . . . . . . 12 |- (C e. NN -> C e. NN)
25		id 73	. . . . . . . . . . . 12 |- (B e. NN -> B e. NN)
26	23, 24, 25	3anim123i 1053	. . . . . . . . . . 11 |- ((A e. RR /\ C e. NN /\ B e. NN) -> (A e. CC /\ C e. NN /\ B e. NN))
27	26	3expa 1067	. . . . . . . . . 10 |- (((A e. RR /\ C e. NN) /\ B e. NN) -> (A e. CC /\ C e. NN /\ B e. NN))
28	27	an1rs 547	. . . . . . . . 9 |- (((A e. RR /\ B e. NN) /\ C e. NN) -> (A e. CC /\ C e. NN /\ B e. NN))
29	28	adantllr 433	. . . . . . . 8 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ C e. NN) -> (A e. CC /\ C e. NN /\ B e. NN))
30	29	adantrr 431	. . . . . . 7 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ (C e. NN /\ B < C)) -> (A e. CC /\ C e. NN /\ B e. NN))
31		gt0ne0 6800	. . . . . . . . 9 |- ((A e. RR /\ 0 < A) -> A =/= 0)
32	31	adantr 425	. . . . . . . 8 |- (((A e. RR /\ 0 < A) /\ B e. NN) -> A =/= 0)
33		simpr 350	. . . . . . . 8 |- ((C e. NN /\ B < C) -> B < C)
34	32, 33	anim12i 360	. . . . . . 7 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ (C e. NN /\ B < C)) -> (A =/= 0 /\ B < C))
35		id 73	. . . . . . . . . . 11 |- ((A e. CC /\ A =/= 0) -> (A e. CC /\ A =/= 0))
36	35	3ad2antl1 1038	. . . . . . . . . 10 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ A =/= 0) -> (A e. CC /\ A =/= 0))
37		nnnn0 7315	. . . . . . . . . . . 12 |- (C e. NN -> C e. NN0)
38	37	3ad2ant2 898	. . . . . . . . . . 11 |- ((A e. CC /\ C e. NN /\ B e. NN) -> C e. NN0)
39	38	adantr 425	. . . . . . . . . 10 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ A =/= 0) -> C e. NN0)
40		nnnn0 7315	. . . . . . . . . . . 12 |- (B e. NN -> B e. NN0)
41	40	3ad2ant3 899	. . . . . . . . . . 11 |- ((A e. CC /\ C e. NN /\ B e. NN) -> B e. NN0)
42	41	adantr 425	. . . . . . . . . 10 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ A =/= 0) -> B e. NN0)
43	36, 39, 42	3jca 1050	. . . . . . . . 9 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ A =/= 0) -> ((A e. CC /\ A =/= 0) /\ C e. NN0 /\ B e. NN0))
44	43	adantrr 431	. . . . . . . 8 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ (A =/= 0 /\ B < C)) -> ((A e. CC /\ A =/= 0) /\ C e. NN0 /\ B e. NN0))
45		ltle 6690	. . . . . . . . . . . . 13 |- ((B e. RR /\ C e. RR) -> (B < C -> B <_ C))
46	45	ancoms 484	. . . . . . . . . . . 12 |- ((C e. RR /\ B e. RR) -> (B < C -> B <_ C))
47		nnre 7112	. . . . . . . . . . . 12 |- (C e. NN -> C e. RR)
48		nnre 7112	. . . . . . . . . . . 12 |- (B e. NN -> B e. RR)
49	46, 47, 48	syl2an 503	. . . . . . . . . . 11 |- ((C e. NN /\ B e. NN) -> (B < C -> B <_ C))
50	49	imp 377	. . . . . . . . . 10 |- (((C e. NN /\ B e. NN) /\ B < C) -> B <_ C)
51	50	3adantl1 1032	. . . . . . . . 9 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ B < C) -> B <_ C)
52	51	adantrl 430	. . . . . . . 8 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ (A =/= 0 /\ B < C)) -> B <_ C)
53		expsub 7841	. . . . . . . 8 |- ((((A e. CC /\ A =/= 0) /\ C e. NN0 /\ B e. NN0) /\ B <_ C) -> (A^(C - B)) = ((A^C) / (A^B)))
54	44, 52, 53	syl11anc 524	. . . . . . 7 |- (((A e. CC /\ C e. NN /\ B e. NN) /\ (A =/= 0 /\ B < C)) -> (A^(C - B)) = ((A^C) / (A^B)))
55	30, 34, 54	syl11anc 524	. . . . . 6 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ (C e. NN /\ B < C)) -> (A^(C - B)) = ((A^C) / (A^B)))
56	55	opreq1d 4897	. . . . 5 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ (C e. NN /\ B < C)) -> ((A^(C - B)) x. (F` B)) = (((A^C) / (A^B)) x. (F` B)))
57	2	ffvelrni 4788	. . . . . . . . 9 |- (B e. NN -> (F` B) e. RR)
58	57	recnd 6468	. . . . . . . 8 |- (B e. NN -> (F` B) e. CC)
59	58	ad2antlr 441	. . . . . . 7 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ C e. NN) -> (F` B) e. CC)
60		reexpcl 7823	. . . . . . . . . . 11 |- ((A e. RR /\ B e. NN0) -> (A^B) e. RR)
61	60, 40	sylan2 500	. . . . . . . . . 10 |- ((A e. RR /\ B e. NN) -> (A^B) e. RR)
62	61	recnd 6468	. . . . . . . . 9 |- ((A e. RR /\ B e. NN) -> (A^B) e. CC)
63	62	adantlr 429	. . . . . . . 8 |- (((A e. RR /\ 0 < A) /\ B e. NN) -> (A^B) e. CC)
64	63	adantr 425	. . . . . . 7 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ C e. NN) -> (A^B) e. CC)
65	61	adantlr 429	. . . . . . . . 9 |- (((A e. RR /\ 0 < A) /\ B e. NN) -> (A^B) e. RR)
66		expgt0 7831	. . . . . . . . . . . 12 |- ((A e. RR /\ B e. NN0 /\ 0 < A) -> 0 < (A^B))
67	66, 40	syl3an2 1131	. . . . . . . . . . 11 |- ((A e. RR /\ B e. NN /\ 0 < A) -> 0 < (A^B))
68	67	3expa 1067	. . . . . . . . . 10 |- (((A e. RR /\ B e. NN) /\ 0 < A) -> 0 < (A^B))
69	68	an1rs 547	. . . . . . . . 9 |- (((A e. RR /\ 0 < A) /\ B e. NN) -> 0 < (A^B))
70		gt0ne0 6800	. . . . . . . . 9 |- (((A^B) e. RR /\ 0 < (A^B)) -> (A^B) =/= 0)
71	65, 69, 70	syl11anc 524	. . . . . . . 8 |- (((A e. RR /\ 0 < A) /\ B e. NN) -> (A^B) =/= 0)
72	71	adantr 425	. . . . . . 7 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ C e. NN) -> (A^B) =/= 0)
73		reexpcl 7823	. . . . . . . . . . 11 |- ((A e. RR /\ C e. NN0) -> (A^C) e. RR)
74	73, 37	sylan2 500	. . . . . . . . . 10 |- ((A e. RR /\ C e. NN) -> (A^C) e. RR)
75	74	recnd 6468	. . . . . . . . 9 |- ((A e. RR /\ C e. NN) -> (A^C) e. CC)
76	75	adantlr 429	. . . . . . . 8 |- (((A e. RR /\ 0 < A) /\ C e. NN) -> (A^C) e. CC)
77	76	adantlr 429	. . . . . . 7 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ C e. NN) -> (A^C) e. CC)
78		div13 6926	. . . . . . 7 |- (((F` B) e. CC /\ ((A^B) e. CC /\ (A^B) =/= 0) /\ (A^C) e. CC) -> (((F` B) / (A^B)) x. (A^C)) = (((A^C) / (A^B)) x. (F` B)))
79	59, 64, 72, 77, 78	syl121anc 1105	. . . . . 6 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ C e. NN) -> (((F` B) / (A^B)) x. (A^C)) = (((A^C) / (A^B)) x. (F` B)))
80	79	adantrr 431	. . . . 5 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ (C e. NN /\ B < C)) -> (((F` B) / (A^B)) x. (A^C)) = (((A^C) / (A^B)) x. (F` B)))
81	56, 80	eqtr4d 1928	. . . 4 |- ((((A e. RR /\ 0 < A) /\ B e. NN) /\ (C e. NN /\ B < C)) -> ((A^(C - B)) x. (F` B)) = (((F` B) / (A^B)) x. (A^C)))
82	81	adantlrr 435	. . 3 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) /\ (C e. NN /\ B < C)) -> ((A^(C - B)) x. (F` B)) = (((F` B) / (A^B)) x. (A^C)))
83	22, 82	breqtrd 3361	. 2 |- ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) /\ (C e. NN /\ B < C)) -> (F` C) <_ (((F` B) / (A^B)) x. (A^C)))
84	83	ex 402	1 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> ((C e. NN /\ B < C) -> (F` C) <_ (((F` B) / (A^B)) x. (A^C))))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  ^cexp 7811

This theorem is referenced by:  cvgratlem3 8514

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-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-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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