Theorem tgoldbachgtde 30738

Description: Lemma for tgoldbachgtd 30740. (Contributed by Thierry Arnoux, 15-Dec-2021.)

Hypotheses

Ref	Expression
tgoldbachgtda.o	|- O = { z e. ZZ | -. 2 || z }
tgoldbachgtda.n	|- ( ph -> N e. O )
tgoldbachgtda.0	|- ( ph -> (; 1 0 ^; 2 7 ) <_ N )
tgoldbachgtda.h	|- ( ph -> H : NN --> ( 0 [,) +oo ) )
tgoldbachgtda.k	|- ( ph -> K : NN --> ( 0 [,) +oo ) )
tgoldbachgtda.1	|- ( ( ph /\ m e. NN ) -> ( K ` m ) <_ ( 1 period_ 0_ 7_ 9_ 9_ 5 5 ) )
tgoldbachgtda.2	|- ( ( ph /\ m e. NN ) -> ( H ` m ) <_ ( 1 period_ 4_ 1 4 ) )
tgoldbachgtda.3	|- ( ph -> ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( N ^ 2 ) ) <_ S. ( 0 (,) 1 ) ( ( ( ( (Λ oF x. H )vts N ) ` x ) x. ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u N x. x ) ) ) ) _d x )

Assertion

Ref	Expression
tgoldbachgtde	|- ( ph -> 0 < sum_ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )

Distinct variable groups:   m, H, n, x   m, K, n, x   m, N, n, x, z   m, O, n, z   ph, m, n, x

Allowed substitution hints:   ph( z)   H( z)   K( z)   O( x)

Proof of Theorem tgoldbachgtde

Step	Hyp	Ref	Expression
1		tgoldbachgtda.o	. . . . . . . . 9 |- O = { z e. ZZ | -. 2 || z }
2		tgoldbachgtda.n	. . . . . . . . 9 |- ( ph -> N e. O )
3		tgoldbachgtda.0	. . . . . . . . 9 |- ( ph -> (; 1 0 ^; 2 7 ) <_ N )
4	1, 2, 3	tgoldbachgnn 30737	. . . . . . . 8 |- ( ph -> N e. NN )
5	4	nnnn0d 11351	. . . . . . 7 |- ( ph -> N e. NN0 )
6		3nn0 11310	. . . . . . . 8 |- 3 e. NN0
7	6	a1i 11	. . . . . . 7 |- ( ph -> 3 e. NN0 )
8		ssid 3624	. . . . . . . 8 |- NN C_ NN
9	8	a1i 11	. . . . . . 7 |- ( ph -> NN C_ NN )
10	5, 7, 9	reprfi2 30701	. . . . . 6 |- ( ph -> ( NN (repr ` 3 ) N ) e. Fin )
11		diffi 8192	. . . . . 6 |- ( ( NN (repr ` 3 ) N ) e. Fin -> ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) e. Fin )
12	10, 11	syl 17	. . . . 5 |- ( ph -> ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) e. Fin )
13		difssd 3738	. . . . . . 7 |- ( ph -> ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) C_ ( NN (repr ` 3 ) N ) )
14	13	sselda 3603	. . . . . 6 |- ( ( ph /\ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) -> n e. ( NN (repr ` 3 ) N ) )
15		vmaf 24845	. . . . . . . . . 10 |- Λ : NN --> RR
16	15	a1i 11	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> Λ : NN --> RR )
17	8	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> NN C_ NN )
18	5	nn0zd 11480	. . . . . . . . . . . 12 |- ( ph -> N e. ZZ )
19	18	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> N e. ZZ )
20	6	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 3 e. NN0 )
21		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> n e. ( NN (repr ` 3 ) N ) )
22	17, 19, 20, 21	reprf 30690	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> n : ( 0..^ 3 ) --> NN )
23		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
24	23	tpid1 4303	. . . . . . . . . . . 12 |- 0 e. { 0 , 1 , 2 }
25		fzo0to3tp 12554	. . . . . . . . . . . 12 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
26	24, 25	eleqtrri 2700	. . . . . . . . . . 11 |- 0 e. ( 0..^ 3 )
27	26	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 0 e. ( 0..^ 3 ) )
28	22, 27	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( n ` 0 ) e. NN )
29	16, 28	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> (Λ ` ( n ` 0 ) ) e. RR )
30		tgoldbachgtda.h	. . . . . . . . . . 11 |- ( ph -> H : NN --> ( 0 [,) +oo ) )
31		rge0ssre 12280	. . . . . . . . . . 11 |- ( 0 [,) +oo ) C_ RR
32		fss 6056	. . . . . . . . . . 11 |- ( ( H : NN --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ RR ) -> H : NN --> RR )
33	30, 31, 32	sylancl 694	. . . . . . . . . 10 |- ( ph -> H : NN --> RR )
34	33	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> H : NN --> RR )
35	34, 28	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( H ` ( n ` 0 ) ) e. RR )
36	29, 35	remulcld 10070	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) e. RR )
37		1ex 10035	. . . . . . . . . . . . . 14 |- 1 e. _V
38	37	tpid2 4304	. . . . . . . . . . . . 13 |- 1 e. { 0 , 1 , 2 }
39	38, 25	eleqtrri 2700	. . . . . . . . . . . 12 |- 1 e. ( 0..^ 3 )
40	39	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 1 e. ( 0..^ 3 ) )
41	22, 40	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( n ` 1 ) e. NN )
42	16, 41	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> (Λ ` ( n ` 1 ) ) e. RR )
43		tgoldbachgtda.k	. . . . . . . . . . . 12 |- ( ph -> K : NN --> ( 0 [,) +oo ) )
44		fss 6056	. . . . . . . . . . . 12 |- ( ( K : NN --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ RR ) -> K : NN --> RR )
45	43, 31, 44	sylancl 694	. . . . . . . . . . 11 |- ( ph -> K : NN --> RR )
46	45	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> K : NN --> RR )
47	46, 41	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( K ` ( n ` 1 ) ) e. RR )
48	42, 47	remulcld 10070	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) e. RR )
49		2ex 11092	. . . . . . . . . . . . . 14 |- 2 e. _V
50	49	tpid3 4307	. . . . . . . . . . . . 13 |- 2 e. { 0 , 1 , 2 }
51	50, 25	eleqtrri 2700	. . . . . . . . . . . 12 |- 2 e. ( 0..^ 3 )
52	51	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 2 e. ( 0..^ 3 ) )
53	22, 52	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( n ` 2 ) e. NN )
54	16, 53	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> (Λ ` ( n ` 2 ) ) e. RR )
55	46, 53	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( K ` ( n ` 2 ) ) e. RR )
56	54, 55	remulcld 10070	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) e. RR )
57	48, 56	remulcld 10070	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) e. RR )
58	36, 57	remulcld 10070	. . . . . 6 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. RR )
59	14, 58	syldan 487	. . . . 5 |- ( ( ph /\ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) -> ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. RR )
60	12, 59	fsumrecl 14465	. . . 4 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. RR )
61		0nn0 11307	. . . . . . 7 |- 0 e. NN0
62		qssre 11798	. . . . . . . 8 |- QQ C_ RR
63		4nn0 11311	. . . . . . . . . . . 12 |- 4 e. NN0
64		2nn0 11309	. . . . . . . . . . . . 13 |- 2 e. NN0
65		nn0ssq 11796	. . . . . . . . . . . . . . . 16 |- NN0 C_ QQ
66		8nn0 11315	. . . . . . . . . . . . . . . 16 |- 8 e. NN0
67	65, 66	sselii 3600	. . . . . . . . . . . . . . 15 |- 8 e. QQ
68	63, 67	dp2clq 29588	. . . . . . . . . . . . . 14 |- _ 4 8 e. QQ
69	64, 68	dp2clq 29588	. . . . . . . . . . . . 13 |- _ 2_ 4 8 e. QQ
70	64, 69	dp2clq 29588	. . . . . . . . . . . 12 |- _ 2_ 2_ 4 8 e. QQ
71	63, 70	dp2clq 29588	. . . . . . . . . . 11 |- _ 4_ 2_ 2_ 4 8 e. QQ
72	61, 71	dp2clq 29588	. . . . . . . . . 10 |- _ 0_ 4_ 2_ 2_ 4 8 e. QQ
73	61, 72	dp2clq 29588	. . . . . . . . 9 |- _ 0_ 0_ 4_ 2_ 2_ 4 8 e. QQ
74	61, 73	dp2clq 29588	. . . . . . . 8 |- _ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 e. QQ
75	62, 74	sselii 3600	. . . . . . 7 |- _ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 e. RR
76		dpcl 29598	. . . . . . 7 |- ( ( 0 e. NN0 /\ _ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 e. RR ) -> ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) e. RR )
77	61, 75, 76	mp2an 708	. . . . . 6 |- ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) e. RR
78	77	a1i 11	. . . . 5 |- ( ph -> ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) e. RR )
79	4	nnred 11035	. . . . . 6 |- ( ph -> N e. RR )
80	79	resqcld 13035	. . . . 5 |- ( ph -> ( N ^ 2 ) e. RR )
81	78, 80	remulcld 10070	. . . 4 |- ( ph -> ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( N ^ 2 ) ) e. RR )
82	10, 58	fsumrecl 14465	. . . 4 |- ( ph -> sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. RR )
83		7nn0 11314	. . . . . . . . 9 |- 7 e. NN0
84	6, 68	dp2clq 29588	. . . . . . . . . 10 |- _ 3_ 4 8 e. QQ
85	62, 84	sselii 3600	. . . . . . . . 9 |- _ 3_ 4 8 e. RR
86		dpcl 29598	. . . . . . . . 9 |- ( ( 7 e. NN0 /\ _ 3_ 4 8 e. RR ) -> ( 7 period_ 3_ 4 8 ) e. RR )
87	83, 85, 86	mp2an 708	. . . . . . . 8 |- ( 7 period_ 3_ 4 8 ) e. RR
88	87	a1i 11	. . . . . . 7 |- ( ph -> ( 7 period_ 3_ 4 8 ) e. RR )
89	4	nnrpd 11870	. . . . . . . . 9 |- ( ph -> N e. RR+ )
90	89	relogcld 24369	. . . . . . . 8 |- ( ph -> ( log ` N ) e. RR )
91	5	nn0ge0d 11354	. . . . . . . . 9 |- ( ph -> 0 <_ N )
92	79, 91	resqrtcld 14156	. . . . . . . 8 |- ( ph -> ( sqr ` N ) e. RR )
93	89	sqrtgt0d 14151	. . . . . . . . 9 |- ( ph -> 0 < ( sqr ` N ) )
94	93	gt0ne0d 10592	. . . . . . . 8 |- ( ph -> ( sqr ` N ) =/= 0 )
95	90, 92, 94	redivcld 10853	. . . . . . 7 |- ( ph -> ( ( log ` N ) / ( sqr ` N ) ) e. RR )
96	88, 95	remulcld 10070	. . . . . 6 |- ( ph -> ( ( 7 period_ 3_ 4 8 ) x. ( ( log ` N ) / ( sqr ` N ) ) ) e. RR )
97	96, 80	remulcld 10070	. . . . 5 |- ( ph -> ( ( ( 7 period_ 3_ 4 8 ) x. ( ( log ` N ) / ( sqr ` N ) ) ) x. ( N ^ 2 ) ) e. RR )
98		tgoldbachgtda.1	. . . . . 6 |- ( ( ph /\ m e. NN ) -> ( K ` m ) <_ ( 1 period_ 0_ 7_ 9_ 9_ 5 5 ) )
99		tgoldbachgtda.2	. . . . . 6 |- ( ( ph /\ m e. NN ) -> ( H ` m ) <_ ( 1 period_ 4_ 1 4 ) )
100	1, 4, 3, 30, 43, 98, 99	hgt750leme 30736	. . . . 5 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) <_ ( ( ( 7 period_ 3_ 4 8 ) x. ( ( log ` N ) / ( sqr ` N ) ) ) x. ( N ^ 2 ) ) )
101		2z 11409	. . . . . . . 8 |- 2 e. ZZ
102	101	a1i 11	. . . . . . 7 |- ( ph -> 2 e. ZZ )
103	89, 102	rpexpcld 13032	. . . . . 6 |- ( ph -> ( N ^ 2 ) e. RR+ )
104		hgt750lem 30729	. . . . . . 7 |- ( ( N e. NN0 /\ (; 1 0 ^; 2 7 ) <_ N ) -> ( ( 7 period_ 3_ 4 8 ) x. ( ( log ` N ) / ( sqr ` N ) ) ) < ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) )
105	5, 3, 104	syl2anc 693	. . . . . 6 |- ( ph -> ( ( 7 period_ 3_ 4 8 ) x. ( ( log ` N ) / ( sqr ` N ) ) ) < ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) )
106	96, 78, 103, 105	ltmul1dd 11927	. . . . 5 |- ( ph -> ( ( ( 7 period_ 3_ 4 8 ) x. ( ( log ` N ) / ( sqr ` N ) ) ) x. ( N ^ 2 ) ) < ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( N ^ 2 ) ) )
107	60, 97, 81, 100, 106	lelttrd 10195	. . . 4 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) < ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( N ^ 2 ) ) )
108		tgoldbachgtda.3	. . . . 5 |- ( ph -> ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( N ^ 2 ) ) <_ S. ( 0 (,) 1 ) ( ( ( ( (Λ oF x. H )vts N ) ` x ) x. ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u N x. x ) ) ) ) _d x )
109	33, 45, 5	circlemethhgt 30721	. . . . 5 |- ( ph -> sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) = S. ( 0 (,) 1 ) ( ( ( ( (Λ oF x. H )vts N ) ` x ) x. ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u N x. x ) ) ) ) _d x )
110	108, 109	breqtrrd 4681	. . . 4 |- ( ph -> ( ( 0 period_ 0_ 0_ 0_ 4_ 2_ 2_ 4 8 ) x. ( N ^ 2 ) ) <_ sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )
111	60, 81, 82, 107, 110	ltletrd 10197	. . 3 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) < sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )
112	60, 82	posdifd 10614	. . 3 |- ( ph -> ( sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) < sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) <-> 0 < ( sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) - sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) ) ) )
113	111, 112	mpbid 222	. 2 |- ( ph -> 0 < ( sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) - sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) ) )
114		inss2 3834	. . . . . . . 8 |- ( O i^i Prime ) C_ Prime
115		prmssnn 15390	. . . . . . . 8 |- Prime C_ NN
116	114, 115	sstri 3612	. . . . . . 7 |- ( O i^i Prime ) C_ NN
117	116	a1i 11	. . . . . 6 |- ( ph -> ( O i^i Prime ) C_ NN )
118	9, 18, 7, 117	reprss 30695	. . . . 5 |- ( ph -> ( ( O i^i Prime ) (repr ` 3 ) N ) C_ ( NN (repr ` 3 ) N ) )
119	10, 118	ssfid 8183	. . . 4 |- ( ph -> ( ( O i^i Prime ) (repr ` 3 ) N ) e. Fin )
120	118	sselda 3603	. . . . 5 |- ( ( ph /\ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ) -> n e. ( NN (repr ` 3 ) N ) )
121	58	recnd 10068	. . . . 5 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. CC )
122	120, 121	syldan 487	. . . 4 |- ( ( ph /\ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ) -> ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. CC )
123	119, 122	fsumcl 14464	. . 3 |- ( ph -> sum_ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. CC )
124	60	recnd 10068	. . 3 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) e. CC )
125		disjdif 4040	. . . . 5 |- ( ( ( O i^i Prime ) (repr ` 3 ) N ) i^i ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) = (/)
126	125	a1i 11	. . . 4 |- ( ph -> ( ( ( O i^i Prime ) (repr ` 3 ) N ) i^i ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) = (/) )
127		undif 4049	. . . . . 6 |- ( ( ( O i^i Prime ) (repr ` 3 ) N ) C_ ( NN (repr ` 3 ) N ) <-> ( ( ( O i^i Prime ) (repr ` 3 ) N ) u. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) = ( NN (repr ` 3 ) N ) )
128	118, 127	sylib 208	. . . . 5 |- ( ph -> ( ( ( O i^i Prime ) (repr ` 3 ) N ) u. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) = ( NN (repr ` 3 ) N ) )
129	128	eqcomd 2628	. . . 4 |- ( ph -> ( NN (repr ` 3 ) N ) = ( ( ( O i^i Prime ) (repr ` 3 ) N ) u. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ) )
130	126, 129, 10, 121	fsumsplit 14471	. . 3 |- ( ph -> sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) = ( sum_ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) + sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) ) )
131	123, 124, 130	mvrraddd 10445	. 2 |- ( ph -> ( sum_ n e. ( NN (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) - sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) ) = sum_ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )
132	113, 131	breqtrd 4679	1 |- ( ph -> 0 < sum_ n e. ( ( O i^i Prime ) (repr ` 3 ) N ) ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {ctp 4181   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   7c7 11075   8c8 11076   9c9 11077   NN0cn0 11292   ZZcz 11377  ;cdc 11493   QQcq 11788   (,)cioo 12175   [,)cico 12177  ..^cfzo 12465   ^cexp 12860   sqrcsqrt 13973   sum_csu 14416   expce 14792   picpi 14797   || cdvds 14983   Primecprime 15385   S.citg 23387   logclog 24301  Λcvma 24818  _cdp2 29577   periodcdp 29595  reprcrepr 30686  vtscvts 30713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-ros335 30723  ax-ros336 30724

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-r1 8627  df-rank 8628  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-pmtr 17862  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-limc 23630  df-dv 23631  df-ulm 24131  df-log 24303  df-cxp 24304  df-atan 24594  df-cht 24823  df-vma 24824  df-chp 24825  df-dp2 29578  df-dp 29596  df-repr 30687  df-vts 30714

This theorem is referenced by:  tgoldbachgtda  30739