Theorem cvgratlem1 8512

Description: Lemma for cvgrati 8517. Establish, by induction, an exponential upper bound for the terms of a real series, given that the ratio of successive terms is less than some positive constant A beyond a starting index B.

Hypothesis

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

Assertion

Ref	Expression
cvgratlem1	|- ((C 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)) <_ ((A^C) x. (F` B)))

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

Proof of Theorem cvgratlem1

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . 7 |- (y = 1 -> (B + y) = (B + 1))
2	1	fveq2d 4685	. . . . . 6 |- (y = 1 -> (F` (B + y)) = (F` (B + 1)))
3		opreq2 4890	. . . . . . 7 |- (y = 1 -> (A^y) = (A^1))
4	3	opreq1d 4897	. . . . . 6 |- (y = 1 -> ((A^y) x. (F` B)) = ((A^1) x. (F` B)))
5	2, 4	breq12d 3351	. . . . 5 |- (y = 1 -> ((F` (B + y)) < ((A^y) x. (F` B)) <-> (F` (B + 1)) < ((A^1) x. (F` B))))
6	5	imbi2d 674	. . . 4 |- (y = 1 -> ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + y)) < ((A^y) x. (F` B))) <-> (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + 1)) < ((A^1) x. (F` B)))))
7		opreq2 4890	. . . . . . 7 |- (y = z -> (B + y) = (B + z))
8	7	fveq2d 4685	. . . . . 6 |- (y = z -> (F` (B + y)) = (F` (B + z)))
9		opreq2 4890	. . . . . . 7 |- (y = z -> (A^y) = (A^z))
10	9	opreq1d 4897	. . . . . 6 |- (y = z -> ((A^y) x. (F` B)) = ((A^z) x. (F` B)))
11	8, 10	breq12d 3351	. . . . 5 |- (y = z -> ((F` (B + y)) < ((A^y) x. (F` B)) <-> (F` (B + z)) < ((A^z) x. (F` B))))
12	11	imbi2d 674	. . . 4 |- (y = z -> ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + y)) < ((A^y) x. (F` B))) <-> (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + z)) < ((A^z) x. (F` B)))))
13		opreq2 4890	. . . . . . 7 |- (y = (z + 1) -> (B + y) = (B + (z + 1)))
14	13	fveq2d 4685	. . . . . 6 |- (y = (z + 1) -> (F` (B + y)) = (F` (B + (z + 1))))
15		opreq2 4890	. . . . . . 7 |- (y = (z + 1) -> (A^y) = (A^(z + 1)))
16	15	opreq1d 4897	. . . . . 6 |- (y = (z + 1) -> ((A^y) x. (F` B)) = ((A^(z + 1)) x. (F` B)))
17	14, 16	breq12d 3351	. . . . 5 |- (y = (z + 1) -> ((F` (B + y)) < ((A^y) x. (F` B)) <-> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
18	17	imbi2d 674	. . . 4 |- (y = (z + 1) -> ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + y)) < ((A^y) x. (F` B))) <-> (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B)))))
19		opreq2 4890	. . . . . . 7 |- (y = C -> (B + y) = (B + C))
20	19	fveq2d 4685	. . . . . 6 |- (y = C -> (F` (B + y)) = (F` (B + C)))
21		opreq2 4890	. . . . . . 7 |- (y = C -> (A^y) = (A^C))
22	21	opreq1d 4897	. . . . . 6 |- (y = C -> ((A^y) x. (F` B)) = ((A^C) x. (F` B)))
23	20, 22	breq12d 3351	. . . . 5 |- (y = C -> ((F` (B + y)) < ((A^y) x. (F` B)) <-> (F` (B + C)) < ((A^C) x. (F` B))))
24	23	imbi2d 674	. . . 4 |- (y = C -> ((((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + y)) < ((A^y) x. (F` B))) <-> (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + C)) < ((A^C) x. (F` B)))))
25		nnre 7112	. . . . . . . . 9 |- (B e. NN -> B e. RR)
26		leid 6701	. . . . . . . . 9 |- (B e. RR -> B <_ B)
27	25, 26	syl 12	. . . . . . . 8 |- (B e. NN -> B <_ B)
28		breq2 3342	. . . . . . . . . 10 |- (x = B -> (B <_ x <-> B <_ B))
29		opreq1 4889	. . . . . . . . . . . 12 |- (x = B -> (x + 1) = (B + 1))
30	29	fveq2d 4685	. . . . . . . . . . 11 |- (x = B -> (F` (x + 1)) = (F` (B + 1)))
31		fveq2 4681	. . . . . . . . . . . 12 |- (x = B -> (F` x) = (F` B))
32	31	opreq2d 4898	. . . . . . . . . . 11 |- (x = B -> (A x. (F` x)) = (A x. (F` B)))
33	30, 32	breq12d 3351	. . . . . . . . . 10 |- (x = B -> ((F` (x + 1)) < (A x. (F` x)) <-> (F` (B + 1)) < (A x. (F` B))))
34	28, 33	imbi12d 688	. . . . . . . . 9 |- (x = B -> ((B <_ x -> (F` (x + 1)) < (A x. (F` x))) <-> (B <_ B -> (F` (B + 1)) < (A x. (F` B)))))
35	34	rcla4v 2376	. . . . . . . 8 |- (B e. NN -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (B <_ B -> (F` (B + 1)) < (A x. (F` B)))))
36	27, 35	mpid 58	. . . . . . 7 |- (B e. NN -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` (B + 1)) < (A x. (F` B))))
37	36	imp 377	. . . . . 6 |- ((B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))) -> (F` (B + 1)) < (A x. (F` B)))
38	37	adantl 424	. . . . 5 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + 1)) < (A x. (F` B)))
39		recn 6466	. . . . . . . . 9 |- (A e. RR -> A e. CC)
40		exp1 7816	. . . . . . . . 9 |- (A e. CC -> (A^1) = A)
41	39, 40	syl 12	. . . . . . . 8 |- (A e. RR -> (A^1) = A)
42	41	adantr 425	. . . . . . 7 |- ((A e. RR /\ 0 < A) -> (A^1) = A)
43	42	opreq1d 4897	. . . . . 6 |- ((A e. RR /\ 0 < A) -> ((A^1) x. (F` B)) = (A x. (F` B)))
44	43	adantr 425	. . . . 5 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> ((A^1) x. (F` B)) = (A x. (F` B)))
45	38, 44	breqtrrd 3363	. . . 4 |- (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + 1)) < ((A^1) x. (F` B)))
46		nnaddcl 7123	. . . . . . . . . . . . . . . . . . 19 |- ((B e. NN /\ z e. NN) -> (B + z) e. NN)
47	46	ancoms 484	. . . . . . . . . . . . . . . . . 18 |- ((z e. NN /\ B e. NN) -> (B + z) e. NN)
48		peano2nn 7118	. . . . . . . . . . . . . . . . . 18 |- ((B + z) e. NN -> ((B + z) + 1) e. NN)
49	47, 48	syl 12	. . . . . . . . . . . . . . . . 17 |- ((z e. NN /\ B e. NN) -> ((B + z) + 1) e. NN)
50		cvgratlem1.1	. . . . . . . . . . . . . . . . . 18 |- F:NN-->RR
51	50	ffvelrni 4788	. . . . . . . . . . . . . . . . 17 |- (((B + z) + 1) e. NN -> (F` ((B + z) + 1)) e. RR)
52	49, 51	syl 12	. . . . . . . . . . . . . . . 16 |- ((z e. NN /\ B e. NN) -> (F` ((B + z) + 1)) e. RR)
53	52	adantll 428	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (F` ((B + z) + 1)) e. RR)
54		remulcl 6457	. . . . . . . . . . . . . . . . 17 |- ((A e. RR /\ (F` (B + z)) e. RR) -> (A x. (F` (B + z))) e. RR)
55	50	ffvelrni 4788	. . . . . . . . . . . . . . . . . 18 |- ((B + z) e. NN -> (F` (B + z)) e. RR)
56	47, 55	syl 12	. . . . . . . . . . . . . . . . 17 |- ((z e. NN /\ B e. NN) -> (F` (B + z)) e. RR)
57	54, 56	sylan2 500	. . . . . . . . . . . . . . . 16 |- ((A e. RR /\ (z e. NN /\ B e. NN)) -> (A x. (F` (B + z))) e. RR)
58	57	anassrs 489	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (A x. (F` (B + z))) e. RR)
59		simpll 448	. . . . . . . . . . . . . . . 16 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> A e. RR)
60		remulcl 6457	. . . . . . . . . . . . . . . . 17 |- (((A^z) e. RR /\ (F` B) e. RR) -> ((A^z) x. (F` B)) e. RR)
61		reexpcl 7823	. . . . . . . . . . . . . . . . . 18 |- ((A e. RR /\ z e. NN0) -> (A^z) e. RR)
62		nnnn0 7315	. . . . . . . . . . . . . . . . . 18 |- (z e. NN -> z e. NN0)
63	61, 62	sylan2 500	. . . . . . . . . . . . . . . . 17 |- ((A e. RR /\ z e. NN) -> (A^z) e. RR)
64	50	ffvelrni 4788	. . . . . . . . . . . . . . . . 17 |- (B e. NN -> (F` B) e. RR)
65	60, 63, 64	syl2an 503	. . . . . . . . . . . . . . . 16 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> ((A^z) x. (F` B)) e. RR)
66		remulcl 6457	. . . . . . . . . . . . . . . 16 |- ((A e. RR /\ ((A^z) x. (F` B)) e. RR) -> (A x. ((A^z) x. (F` B))) e. RR)
67	59, 65, 66	syl11anc 524	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (A x. ((A^z) x. (F` B))) e. RR)
68		axlttrn 6673	. . . . . . . . . . . . . . 15 |- (((F` ((B + z) + 1)) e. RR /\ (A x. (F` (B + z))) e. RR /\ (A x. ((A^z) x. (F` B))) e. RR) -> (((F` ((B + z) + 1)) < (A x. (F` (B + z))) /\ (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B)))) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B)))))
69	53, 58, 67, 68	syl111anc 1100	. . . . . . . . . . . . . 14 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (((F` ((B + z) + 1)) < (A x. (F` (B + z))) /\ (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B)))) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B)))))
70		nn0addge1 7339	. . . . . . . . . . . . . . . . . 18 |- ((B e. RR /\ z e. NN0) -> B <_ (B + z))
71	70, 25, 62	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((B e. NN /\ z e. NN) -> B <_ (B + z))
72		breq2 3342	. . . . . . . . . . . . . . . . . . . 20 |- (x = (B + z) -> (B <_ x <-> B <_ (B + z)))
73		opreq1 4889	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = (B + z) -> (x + 1) = ((B + z) + 1))
74	73	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . 21 |- (x = (B + z) -> (F` (x + 1)) = (F` ((B + z) + 1)))
75		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = (B + z) -> (F` x) = (F` (B + z)))
76	75	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . 21 |- (x = (B + z) -> (A x. (F` x)) = (A x. (F` (B + z))))
77	74, 76	breq12d 3351	. . . . . . . . . . . . . . . . . . . 20 |- (x = (B + z) -> ((F` (x + 1)) < (A x. (F` x)) <-> (F` ((B + z) + 1)) < (A x. (F` (B + z)))))
78	72, 77	imbi12d 688	. . . . . . . . . . . . . . . . . . 19 |- (x = (B + z) -> ((B <_ x -> (F` (x + 1)) < (A x. (F` x))) <-> (B <_ (B + z) -> (F` ((B + z) + 1)) < (A x. (F` (B + z))))))
79	78	rcla4v 2376	. . . . . . . . . . . . . . . . . 18 |- ((B + z) e. NN -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (B <_ (B + z) -> (F` ((B + z) + 1)) < (A x. (F` (B + z))))))
80	46, 79	syl 12	. . . . . . . . . . . . . . . . 17 |- ((B e. NN /\ z e. NN) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (B <_ (B + z) -> (F` ((B + z) + 1)) < (A x. (F` (B + z))))))
81	71, 80	mpid 58	. . . . . . . . . . . . . . . 16 |- ((B e. NN /\ z e. NN) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` ((B + z) + 1)) < (A x. (F` (B + z)))))
82	81	ancoms 484	. . . . . . . . . . . . . . 15 |- ((z e. NN /\ B e. NN) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` ((B + z) + 1)) < (A x. (F` (B + z)))))
83	82	adantll 428	. . . . . . . . . . . . . 14 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (F` ((B + z) + 1)) < (A x. (F` (B + z)))))
84	47	adantll 428	. . . . . . . . . . . . . . . . 17 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (B + z) e. NN)
85	84, 55	syl 12	. . . . . . . . . . . . . . . 16 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (F` (B + z)) e. RR)
86		ltmul2OLD 7010	. . . . . . . . . . . . . . . . . 18 |- ((((F` (B + z)) e. RR /\ ((A^z) x. (F` B)) e. RR /\ A e. RR) /\ 0 < A) -> ((F` (B + z)) < ((A^z) x. (F` B)) <-> (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B)))))
87	86	biimpd 170	. . . . . . . . . . . . . . . . 17 |- ((((F` (B + z)) e. RR /\ ((A^z) x. (F` B)) e. RR /\ A e. RR) /\ 0 < A) -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B)))))
88	87	ex 402	. . . . . . . . . . . . . . . 16 |- (((F` (B + z)) e. RR /\ ((A^z) x. (F` B)) e. RR /\ A e. RR) -> (0 < A -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B))))))
89	85, 65, 59, 88	syl111anc 1100	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (0 < A -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B))))))
90	89	imp3a 388	. . . . . . . . . . . . . 14 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> ((0 < A /\ (F` (B + z)) < ((A^z) x. (F` B))) -> (A x. (F` (B + z))) < (A x. ((A^z) x. (F` B)))))
91	69, 83, 90	syl2and 508	. . . . . . . . . . . . 13 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> ((A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) /\ (0 < A /\ (F` (B + z)) < ((A^z) x. (F` B)))) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B)))))
92	91	exp4d 412	. . . . . . . . . . . 12 |- (((A e. RR /\ z e. NN) /\ B e. NN) -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (0 < A -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B)))))))
93	92	exp31 407	. . . . . . . . . . 11 |- (A e. RR -> (z e. NN -> (B e. NN -> (A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))) -> (0 < A -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B)))))))))
94	93	imp4a 391	. . . . . . . . . 10 |- (A e. RR -> (z e. NN -> ((B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))) -> (0 < A -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))))))))
95	94	com12 14	. . . . . . . . 9 |- (z e. NN -> (A e. RR -> ((B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x)))) -> (0 < A -> ((F` (B + z)) < ((A^z) x. (F` B)) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))))))))
96	95	com34 40	. . . . . . . 8 |- (z 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 + z)) < ((A^z) x. (F` B)) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))))))))
97	96	imp44 398	. . . . . . 7 |- ((z 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 + z)) < ((A^z) x. (F` B)) -> (F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B)))))
98		ax1cn 6422	. . . . . . . . . . . . . . . 16 |- 1 e. CC
99		addass 6460	. . . . . . . . . . . . . . . 16 |- ((B e. CC /\ z e. CC /\ 1 e. CC) -> ((B + z) + 1) = (B + (z + 1)))
100	98, 99	mp3an3 1180	. . . . . . . . . . . . . . 15 |- ((B e. CC /\ z e. CC) -> ((B + z) + 1) = (B + (z + 1)))
101		nncn 7113	. . . . . . . . . . . . . . 15 |- (B e. NN -> B e. CC)
102		nncn 7113	. . . . . . . . . . . . . . 15 |- (z e. NN -> z e. CC)
103	100, 101, 102	syl2an 503	. . . . . . . . . . . . . 14 |- ((B e. NN /\ z e. NN) -> ((B + z) + 1) = (B + (z + 1)))
104	103	fveq2d 4685	. . . . . . . . . . . . 13 |- ((B e. NN /\ z e. NN) -> (F` ((B + z) + 1)) = (F` (B + (z + 1))))
105	104	adantll 428	. . . . . . . . . . . 12 |- (((A e. CC /\ B e. NN) /\ z e. NN) -> (F` ((B + z) + 1)) = (F` (B + (z + 1))))
106		expp1 7817	. . . . . . . . . . . . . . . . 17 |- ((A e. CC /\ z e. NN0) -> (A^(z + 1)) = ((A^z) x. A))
107	106, 62	sylan2 500	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ z e. NN) -> (A^(z + 1)) = ((A^z) x. A))
108		expcl 7824	. . . . . . . . . . . . . . . . . 18 |- ((A e. CC /\ z e. NN0) -> (A^z) e. CC)
109	108, 62	sylan2 500	. . . . . . . . . . . . . . . . 17 |- ((A e. CC /\ z e. NN) -> (A^z) e. CC)
110		simpl 346	. . . . . . . . . . . . . . . . 17 |- ((A e. CC /\ z e. NN) -> A e. CC)
111		mulcom 6459	. . . . . . . . . . . . . . . . 17 |- (((A^z) e. CC /\ A e. CC) -> ((A^z) x. A) = (A x. (A^z)))
112	109, 110, 111	syl11anc 524	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ z e. NN) -> ((A^z) x. A) = (A x. (A^z)))
113	107, 112	eqtrd 1925	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ z e. NN) -> (A^(z + 1)) = (A x. (A^z)))
114	113	adantlr 429	. . . . . . . . . . . . . 14 |- (((A e. CC /\ B e. NN) /\ z e. NN) -> (A^(z + 1)) = (A x. (A^z)))
115	114	opreq1d 4897	. . . . . . . . . . . . 13 |- (((A e. CC /\ B e. NN) /\ z e. NN) -> ((A^(z + 1)) x. (F` B)) = ((A x. (A^z)) x. (F` B)))
116		mulass 6461	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ (A^z) e. CC /\ (F` B) e. CC) -> ((A x. (A^z)) x. (F` B)) = (A x. ((A^z) x. (F` B))))
117	116	3expa 1067	. . . . . . . . . . . . . . 15 |- (((A e. CC /\ (A^z) e. CC) /\ (F` B) e. CC) -> ((A x. (A^z)) x. (F` B)) = (A x. ((A^z) x. (F` B))))
118	110, 109	jca 310	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ z e. NN) -> (A e. CC /\ (A^z) e. CC))
119	64	recnd 6468	. . . . . . . . . . . . . . 15 |- (B e. NN -> (F` B) e. CC)
120	117, 118, 119	syl2an 503	. . . . . . . . . . . . . 14 |- (((A e. CC /\ z e. NN) /\ B e. NN) -> ((A x. (A^z)) x. (F` B)) = (A x. ((A^z) x. (F` B))))
121	120	an1rs 547	. . . . . . . . . . . . 13 |- (((A e. CC /\ B e. NN) /\ z e. NN) -> ((A x. (A^z)) x. (F` B)) = (A x. ((A^z) x. (F` B))))
122	115, 121	eqtr2d 1926	. . . . . . . . . . . 12 |- (((A e. CC /\ B e. NN) /\ z e. NN) -> (A x. ((A^z) x. (F` B))) = ((A^(z + 1)) x. (F` B)))
123	105, 122	breq12d 3351	. . . . . . . . . . 11 |- (((A e. CC /\ B e. NN) /\ z e. NN) -> ((F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))) <-> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
124	123, 39	sylanl1 509	. . . . . . . . . 10 |- (((A e. RR /\ B e. NN) /\ z e. NN) -> ((F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))) <-> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
125	124	ancoms 484	. . . . . . . . 9 |- ((z e. NN /\ (A e. RR /\ B e. NN)) -> ((F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))) <-> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
126	125	adantrlr 437	. . . . . . . 8 |- ((z e. NN /\ ((A e. RR /\ 0 < A) /\ B e. NN)) -> ((F` ((B + z) + 1)) < (A x. ((A^z) x. (F` B))) <-> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
127	126	adantrrr 439	. . . . . . 7 |- ((z 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 + z) + 1)) < (A x. ((A^z) x. (F` B))) <-> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
128	97, 127	sylibd 219	. . . . . 6 |- ((z 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 + z)) < ((A^z) x. (F` B)) -> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B))))
129	128	ex 402	. . . . 5 |- (z 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 + z)) < ((A^z) x. (F` B)) -> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B)))))
130	129	a2d 16	. . . 4 |- (z 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 + z)) < ((A^z) x. (F` B))) -> (((A e. RR /\ 0 < A) /\ (B e. NN /\ A.x e. NN (B <_ x -> (F` (x + 1)) < (A x. (F` x))))) -> (F` (B + (z + 1))) < ((A^(z + 1)) x. (F` B)))))
131	6, 12, 18, 24, 45, 130	nnind 7120	. . 3 |- (C 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)) < ((A^C) x. (F` B))))
132	131	imp 377	. 2 |- ((C 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)) < ((A^C) x. (F` B)))
133		nnaddcl 7123	. . . . . . . 8 |- ((B e. NN /\ C e. NN) -> (B + C) e. NN)
134	133	ancoms 484	. . . . . . 7 |- ((C e. NN /\ B e. NN) -> (B + C) e. NN)
135	50	ffvelrni 4788	. . . . . . 7 |- ((B + C) e. NN -> (F` (B + C)) e. RR)
136	134, 135	syl 12	. . . . . 6 |- ((C e. NN /\ B e. NN) -> (F` (B + C)) e. RR)
137	136	adantrl 430	. . . . 5 |- ((C e. NN /\ (A e. RR /\ B e. NN)) -> (F` (B + C)) e. RR)
138		remulcl 6457	. . . . . . 7 |- (((A^C) e. RR /\ (F` B) e. RR) -> ((A^C) x. (F` B)) e. RR)
139		reexpcl 7823	. . . . . . . . 9 |- ((A e. RR /\ C e. NN0) -> (A^C) e. RR)
140		nnnn0 7315	. . . . . . . . 9 |- (C e. NN -> C e. NN0)
141	139, 140	sylan2 500	. . . . . . . 8 |- ((A e. RR /\ C e. NN) -> (A^C) e. RR)
142	141	ancoms 484	. . . . . . 7 |- ((C e. NN /\ A e. RR) -> (A^C) e. RR)
143	138, 142, 64	syl2an 503	. . . . . 6 |- (((C e. NN /\ A e. RR) /\ B e. NN) -> ((A^C) x. (F` B)) e. RR)
144	143	anasss 488	. . . . 5 |- ((C e. NN /\ (A e. RR /\ B e. NN)) -> ((A^C) x. (F` B)) e. RR)
145		ltle 6690	. . . . 5 |- (((F` (B + C)) e. RR /\ ((A^C) x. (F` B)) e. RR) -> ((F` (B + C)) < ((A^C) x. (F` B)) -> (F` (B + C)) <_ ((A^C) x. (F` B))))
146	137, 144, 145	syl11anc 524	. . . 4 |- ((C e. NN /\ (A e. RR /\ B e. NN)) -> ((F` (B + C)) < ((A^C) x. (F` B)) -> (F` (B + C)) <_ ((A^C) x. (F` B))))
147	146	adantrlr 437	. . 3 |- ((C e. NN /\ ((A e. RR /\ 0 < A) /\ B e. NN)) -> ((F` (B + C)) < ((A^C) x. (F` B)) -> (F` (B + C)) <_ ((A^C) x. (F` B))))
148	147	adantrrr 439	. 2 |- ((C 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)) < ((A^C) x. (F` B)) -> (F` (B + C)) <_ ((A^C) x. (F` B))))
149	132, 148	mpd 29	1 |- ((C 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)) <_ ((A^C) x. (F` B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  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   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  ^cexp 7811

This theorem is referenced by:  cvgratlem2 8513

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-n 7108  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

Copyright terms: Public domain