Theorem fsumltisumi 15823

Description: A partial sum of a series with positive terms is less than the infinite sum.

Hypothesis

Ref	Expression
fsumltisumi.1	|- N e. (ZZ>=` M)

Assertion

Ref	Expression
fsumltisumi	|- ((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x) -> sum_k e. (M...N)(F` k) < sum_k e. (ZZ>=` M)(F` k))

Distinct variable groups:   k,M,x   k,N,x   k,F,x

Proof of Theorem fsumltisumi

Step	Hyp	Ref	Expression
1		fveq1 4680	. . . 4 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (F` k) = (if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k))
2	1	sumeq2sdv 8253	. . 3 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> sum_k e. (M...N)(F` k) = sum_k e. (M...N)(if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k))
3	1	sumeq2sdv 8253	. . 3 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> sum_k e. (ZZ>=` M)(F` k) = sum_k e. (ZZ>=` M)(if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k))
4	2, 3	breq12d 3351	. 2 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (sum_k e. (M...N)(F` k) < sum_k e. (ZZ>=` M)(F` k) <-> sum_k e. (M...N)(if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k) < sum_k e. (ZZ>=` M)(if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k)))
5		fsumltisumi.1	. . 3 |- N e. (ZZ>=` M)
6		feq1 4551	. . . . . 6 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (F:(ZZ>=` M)-->RR+ <-> if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}):(ZZ>=` M)-->RR+))
7		ax-17 1317	. . . . . . . 8 |- (u e. F -> A.x u e. F)
8		ax-17 1317	. . . . . . . . . 10 |- (F:(ZZ>=` M)-->RR+ -> A.x F:(ZZ>=` M)-->RR+)
9		hbe1 1363	. . . . . . . . . 10 |- (E.x(<.M, + >. seq F) ~~> x -> A.xE.x(<.M, + >. seq F) ~~> x)
10	8, 9	hban 1356	. . . . . . . . 9 |- ((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x) -> A.x(F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x))
11		ax-17 1317	. . . . . . . . 9 |- (u e. {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} -> A.x u e. {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})
12	10, 7, 11	hbif 2999	. . . . . . . 8 |- (u e. if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> A.x u e. if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}))
13	7, 12	hbeq 1995	. . . . . . 7 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> A.x F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}))
14		opreq2 4890	. . . . . . . 8 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (<.M, + >. seq F) = (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})))
15	14	breq1d 3348	. . . . . . 7 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> ((<.M, + >. seq F) ~~> x <-> (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x))
16	13, 15	exbid 1460	. . . . . 6 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (E.x(<.M, + >. seq F) ~~> x <-> E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x))
17	6, 16	anbi12d 690	. . . . 5 |- (F = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> ((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x) <-> (if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}):(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x)))
18		feq1 4551	. . . . . 6 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}:(ZZ>=` M)-->RR+ <-> if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}):(ZZ>=` M)-->RR+))
19	11, 12	hbeq 1995	. . . . . . 7 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> A.x{<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}))
20		opreq2 4890	. . . . . . . 8 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) = (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})))
21	20	breq1d 3348	. . . . . . 7 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> ((<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x <-> (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x))
22	19, 21	exbid 1460	. . . . . 6 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (E.x(<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x <-> E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x))
23	18, 22	anbi12d 690	. . . . 5 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) -> (({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x) <-> (if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}):(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x)))
24		eqid 1884	. . . . . . 7 |- {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} = {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}
25		rpexpcl 7825	. . . . . . . 8 |- (((1 / 2) e. RR+ /\ (y - M) e. NN0) -> ((1 / 2)^(y - M)) e. RR+)
26		2re 7163	. . . . . . . . . 10 |- 2 e. RR
27		2pos 7173	. . . . . . . . . 10 |- 0 < 2
28	26, 27	elrpii 7234	. . . . . . . . 9 |- 2 e. RR+
29		rpreccl 7250	. . . . . . . . 9 |- (2 e. RR+ -> (1 / 2) e. RR+)
30	28, 29	ax-mp 7	. . . . . . . 8 |- (1 / 2) e. RR+
31		0z 7355	. . . . . . . . . 10 |- 0 e. ZZ
32		eluzel2 7593	. . . . . . . . . . 11 |- (N e. (ZZ>=` M) -> M e. ZZ)
33	5, 32	ax-mp 7	. . . . . . . . . 10 |- M e. ZZ
34	31, 33	eluzsubi 7606	. . . . . . . . 9 |- (y e. (ZZ>=` (0 + M)) -> (y - M) e. (ZZ>=` 0))
35		zcn 7349	. . . . . . . . . . . . . 14 |- (M e. ZZ -> M e. CC)
36	33, 35	ax-mp 7	. . . . . . . . . . . . 13 |- M e. CC
37	36	addid2i 6484	. . . . . . . . . . . 12 |- (0 + M) = M
38	37	eqcomi 1888	. . . . . . . . . . 11 |- M = (0 + M)
39	38	fveq2i 4684	. . . . . . . . . 10 |- (ZZ>=` M) = (ZZ>=` (0 + M))
40	39	eleq2i 1961	. . . . . . . . 9 |- (y e. (ZZ>=` M) <-> y e. (ZZ>=` (0 + M)))
41		elnn0uz 7610	. . . . . . . . 9 |- ((y - M) e. NN0 <-> (y - M) e. (ZZ>=` 0))
42	34, 40, 41	3imtr4i 236	. . . . . . . 8 |- (y e. (ZZ>=` M) -> (y - M) e. NN0)
43	25, 30, 42	sylancr 526	. . . . . . 7 |- (y e. (ZZ>=` M) -> ((1 / 2)^(y - M)) e. RR+)
44	24, 43	fopab 4800	. . . . . 6 |- {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}:(ZZ>=` M)-->RR+
45		addex 6470	. . . . . . . . . 10 |- + e. _V
46		nn0ex 7314	. . . . . . . . . . 11 |- NN0 e. _V
47	46	opabex2 4539	. . . . . . . . . 10 |- {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))} e. _V
48	45, 47	seq0seqz 7785	. . . . . . . . 9 |- ( + seq0 {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}) = (<.0, + >. seq {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))})
49		eqid 1884	. . . . . . . . . 10 |- {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))} = {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}
50		2cn 7164	. . . . . . . . . . 11 |- 2 e. CC
51		2ne0 7174	. . . . . . . . . . 11 |- 2 =/= 0
52	50, 51	reccli 6902	. . . . . . . . . 10 |- (1 / 2) e. CC
53	26, 51	rereccli 6979	. . . . . . . . . . . . 13 |- (1 / 2) e. RR
54	53	absrei 8126	. . . . . . . . . . . 12 |- (abs` (1 / 2)) = (sqr` ((1 / 2)^2))
55	52	sqvali 7859	. . . . . . . . . . . . 13 |- ((1 / 2)^2) = ((1 / 2) x. (1 / 2))
56	55	fveq2i 4684	. . . . . . . . . . . 12 |- (sqr` ((1 / 2)^2)) = (sqr` ((1 / 2) x. (1 / 2)))
57		0re 6603	. . . . . . . . . . . . . . 15 |- 0 e. RR
58		1re 6598	. . . . . . . . . . . . . . 15 |- 1 e. RR
59		lt01 6871	. . . . . . . . . . . . . . 15 |- 0 < 1
60	57, 58, 59	ltleii 6756	. . . . . . . . . . . . . 14 |- 0 <_ 1
61	58, 26	divge0i 7040	. . . . . . . . . . . . . 14 |- ((0 <_ 1 /\ 0 < 2) -> 0 <_ (1 / 2))
62	60, 27, 61	mp2an 761	. . . . . . . . . . . . 13 |- 0 <_ (1 / 2)
63	53	sqrmsqi 7959	. . . . . . . . . . . . 13 |- (0 <_ (1 / 2) -> (sqr` ((1 / 2) x. (1 / 2))) = (1 / 2))
64	62, 63	ax-mp 7	. . . . . . . . . . . 12 |- (sqr` ((1 / 2) x. (1 / 2))) = (1 / 2)
65	54, 56, 64	3eqtri 1912	. . . . . . . . . . 11 |- (abs` (1 / 2)) = (1 / 2)
66		halflt1 7216	. . . . . . . . . . 11 |- (1 / 2) < 1
67	65, 66	eqbrtri 3356	. . . . . . . . . 10 |- (abs` (1 / 2)) < 1
68	49, 52, 67	geolimi 8498	. . . . . . . . 9 |- ( + seq0 {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}) ~~> (1 / (1 - (1 / 2)))
69	48, 68	eqbrtrri 3358	. . . . . . . 8 |- (<.0, + >. seq {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}) ~~> (1 / (1 - (1 / 2)))
70		oprex 4907	. . . . . . . . . 10 |- (1 / (1 - (1 / 2))) e. _V
71		elnn0uz 7610	. . . . . . . . . . . 12 |- (u e. NN0 <-> u e. (ZZ>=` 0))
72		opreq2 4890	. . . . . . . . . . . . . . 15 |- (y = u -> ((1 / 2)^y) = ((1 / 2)^u))
73		oprex 4907	. . . . . . . . . . . . . . 15 |- ((1 / 2)^u) e. _V
74	72, 49, 73	fvopab4 4743	. . . . . . . . . . . . . 14 |- (u e. NN0 -> ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u) = ((1 / 2)^u))
75		expcl 7824	. . . . . . . . . . . . . . 15 |- (((1 / 2) e. CC /\ u e. NN0) -> ((1 / 2)^u) e. CC)
76	52, 75	mpan 759	. . . . . . . . . . . . . 14 |- (u e. NN0 -> ((1 / 2)^u) e. CC)
77	74, 76	eqeltrd 1971	. . . . . . . . . . . . 13 |- (u e. NN0 -> ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u) e. CC)
78		pncan 6557	. . . . . . . . . . . . . . . 16 |- ((u e. CC /\ M e. CC) -> ((u + M) - M) = u)
79		nn0cn 7318	. . . . . . . . . . . . . . . 16 |- (u e. NN0 -> u e. CC)
80	78, 79, 36	sylancl 525	. . . . . . . . . . . . . . 15 |- (u e. NN0 -> ((u + M) - M) = u)
81	80	opreq2d 4898	. . . . . . . . . . . . . 14 |- (u e. NN0 -> ((1 / 2)^((u + M) - M)) = ((1 / 2)^u))
82		addcom 6458	. . . . . . . . . . . . . . . . 17 |- ((u e. CC /\ M e. CC) -> (u + M) = (M + u))
83	82, 79, 36	sylancl 525	. . . . . . . . . . . . . . . 16 |- (u e. NN0 -> (u + M) = (M + u))
84		uzid 7596	. . . . . . . . . . . . . . . . . 18 |- (M e. ZZ -> M e. (ZZ>=` M))
85	33, 84	ax-mp 7	. . . . . . . . . . . . . . . . 17 |- M e. (ZZ>=` M)
86		uzaddcl 7618	. . . . . . . . . . . . . . . . 17 |- ((M e. (ZZ>=` M) /\ u e. NN0) -> (M + u) e. (ZZ>=` M))
87	85, 86	mpan 759	. . . . . . . . . . . . . . . 16 |- (u e. NN0 -> (M + u) e. (ZZ>=` M))
88	83, 87	eqeltrd 1971	. . . . . . . . . . . . . . 15 |- (u e. NN0 -> (u + M) e. (ZZ>=` M))
89		opreq1 4889	. . . . . . . . . . . . . . . . 17 |- (y = (u + M) -> (y - M) = ((u + M) - M))
90	89	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- (y = (u + M) -> ((1 / 2)^(y - M)) = ((1 / 2)^((u + M) - M)))
91		oprex 4907	. . . . . . . . . . . . . . . 16 |- ((1 / 2)^((u + M) - M)) e. _V
92	90, 24, 91	fvopab4 4743	. . . . . . . . . . . . . . 15 |- ((u + M) e. (ZZ>=` M) -> ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ((1 / 2)^((u + M) - M)))
93	88, 92	syl 12	. . . . . . . . . . . . . 14 |- (u e. NN0 -> ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ((1 / 2)^((u + M) - M)))
94	81, 93, 74	3eqtr4d 1937	. . . . . . . . . . . . 13 |- (u e. NN0 -> ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u))
95	77, 94	jca 310	. . . . . . . . . . . 12 |- (u e. NN0 -> (({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u) e. CC /\ ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u)))
96	71, 95	sylbir 218	. . . . . . . . . . 11 |- (u e. (ZZ>=` 0) -> (({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u) e. CC /\ ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u)))
97	96	rgen 2159	. . . . . . . . . 10 |- A.u e. (ZZ>=` 0)(({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u) e. CC /\ ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u))
98		fvex 4689	. . . . . . . . . . . 12 |- (ZZ>=` M) e. _V
99	98	opabex2 4539	. . . . . . . . . . 11 |- {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))} e. _V
100	47, 99, 31, 33	iserzshft2i 8367	. . . . . . . . . 10 |- (((1 / (1 - (1 / 2))) e. _V /\ A.u e. (ZZ>=` 0)(({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u) e. CC /\ ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}` (u + M)) = ({<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}` u))) -> ((<.0, + >. seq {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}) ~~> (1 / (1 - (1 / 2))) <-> (<.(0 + M), + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2)))))
101	70, 97, 100	mp2an 761	. . . . . . . . 9 |- ((<.0, + >. seq {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}) ~~> (1 / (1 - (1 / 2))) <-> (<.(0 + M), + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2))))
102	37	opeq1i 3161	. . . . . . . . . . 11 |- <.(0 + M), + >. = <.M, + >.
103	102	opreq1i 4892	. . . . . . . . . 10 |- (<.(0 + M), + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) = (<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})
104	103	breq1i 3345	. . . . . . . . 9 |- ((<.(0 + M), + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2))) <-> (<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2))))
105	101, 104	bitri 190	. . . . . . . 8 |- ((<.0, + >. seq {<.y, z>. | (y e. NN0 /\ z = ((1 / 2)^y))}) ~~> (1 / (1 - (1 / 2))) <-> (<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2))))
106	69, 105	mpbi 206	. . . . . . 7 |- (<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2)))
107		breq2 3342	. . . . . . . 8 |- (x = (1 / (1 - (1 / 2))) -> ((<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x <-> (<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2)))))
108	70, 107	cla4ev 2371	. . . . . . 7 |- ((<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> (1 / (1 - (1 / 2))) -> E.x(<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x)
109	106, 108	ax-mp 7	. . . . . 6 |- E.x(<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x
110	44, 109	pm3.2i 307	. . . . 5 |- ({<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}) ~~> x)
111	17, 23, 110	elimhyp 3021	. . . 4 |- (if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}):(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x)
112	111	simpli 347	. . 3 |- if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))}):(ZZ>=` M)-->RR+
113	111	simpri 351	. . . 4 |- E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x
114		ax-17 1317	. . . . 5 |- ((<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x -> A.v(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x)
115		ax-17 1317	. . . . . . 7 |- (u e. <.M, + >. -> A.x u e. <.M, + >.)
116		ax-17 1317	. . . . . . 7 |- (u e. seq -> A.x u e. seq )
117	115, 116, 12	hbopr 4904	. . . . . 6 |- (u e. (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) -> A.x u e. (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})))
118		ax-17 1317	. . . . . 6 |- (u e. ~~> -> A.x u e. ~~> )
119		ax-17 1317	. . . . . 6 |- (u e. v -> A.x u e. v)
120	117, 118, 119	hbbr 3381	. . . . 5 |- ((<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> v -> A.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> v)
121		breq2 3342	. . . . 5 |- (x = v -> ((<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x <-> (<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> v))
122	114, 120, 121	cbvex 1529	. . . 4 |- (E.x(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> x <-> E.v(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> v)
123	113, 122	mpbi 206	. . 3 |- E.v(<.M, + >. seq if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})) ~~> v
124	5, 112, 123	fsumltisumii 15822	. 2 |- sum_k e. (M...N)(if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k) < sum_k e. (ZZ>=` M)(if((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x), F, {<.y, z>. | (y e. (ZZ>=` M) /\ z = ((1 / 2)^(y - M)))})` k)
125	4, 124	dedth 3011	1 |- ((F:(ZZ>=` M)-->RR+ /\ E.x(<.M, + >. seq F) ~~> x) -> sum_k e. (M...N)(F` k) < sum_k e. (ZZ>=` M)(F` k))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292  ifcif 2982  <.cop 3046   class class class wbr 3338  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  NN0cn0 6450  ZZcz 6451  RR+crp 6453   < clt 6653  2c2 7145  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774   seq0 cseq0 7775  ^cexp 7811  sqrcsqr 7919  abscabs 8000   ~~> cli 8234  sum_csu 8239

This theorem is referenced by:  fsumltisum 15824

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-sup 5664  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-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240

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