Theorem circlemethhgt 30721

Description: The circle method, where the Vinogradov sums are weighted using the Von Mangoldt function and smoothed using functions H and K. Statement 7.49 of [Helfgott] p. 69. At this point there is no further constraint on the smoothing functions. (Contributed by Thierry Arnoux, 22-Dec-2021.)

Hypotheses

Ref	Expression
circlemethhgt.h	|- ( ph -> H : NN --> RR )
circlemethhgt.k	|- ( ph -> K : NN --> RR )
circlemethhgt.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
circlemethhgt	|- ( 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 )

Distinct variable groups:   n, H, x   n, K, x   n, N, x   ph, n, x

Proof of Theorem circlemethhgt

Dummy variables a y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		circlemethhgt.n	. . 3 |- ( ph -> N e. NN0 )
2		3nn 11186	. . . 4 |- 3 e. NN
3	2	a1i 11	. . 3 |- ( ph -> 3 e. NN )
4		s3len 13639	. . . . . 6 |- ( # ` <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ) = 3
5	4	eqcomi 2631	. . . . 5 |- 3 = ( # ` <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> )
6	5	a1i 11	. . . 4 |- ( ph -> 3 = ( # ` <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ) )
7		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
8		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
9	7, 8	remulcld 10070	. . . . . . . 8 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
10	9	recnd 10068	. . . . . . 7 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. CC )
11		vmaf 24845	. . . . . . . 8 |- Λ : NN --> RR
12	11	a1i 11	. . . . . . 7 |- ( ph -> Λ : NN --> RR )
13		circlemethhgt.h	. . . . . . 7 |- ( ph -> H : NN --> RR )
14		nnex 11026	. . . . . . . 8 |- NN e. _V
15	14	a1i 11	. . . . . . 7 |- ( ph -> NN e. _V )
16		inidm 3822	. . . . . . 7 |- ( NN i^i NN ) = NN
17	10, 12, 13, 15, 15, 16	off 6912	. . . . . 6 |- ( ph -> (Λ oF x. H ) : NN --> CC )
18		cnex 10017	. . . . . . 7 |- CC e. _V
19	18, 14	elmap 7886	. . . . . 6 |- ( (Λ oF x. H ) e. ( CC ^m NN ) <-> (Λ oF x. H ) : NN --> CC )
20	17, 19	sylibr 224	. . . . 5 |- ( ph -> (Λ oF x. H ) e. ( CC ^m NN ) )
21		circlemethhgt.k	. . . . . . 7 |- ( ph -> K : NN --> RR )
22	10, 12, 21, 15, 15, 16	off 6912	. . . . . 6 |- ( ph -> (Λ oF x. K ) : NN --> CC )
23	18, 14	elmap 7886	. . . . . 6 |- ( (Λ oF x. K ) e. ( CC ^m NN ) <-> (Λ oF x. K ) : NN --> CC )
24	22, 23	sylibr 224	. . . . 5 |- ( ph -> (Λ oF x. K ) e. ( CC ^m NN ) )
25	20, 24, 24	s3cld 13617	. . . 4 |- ( ph -> <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> e. Word ( CC ^m NN ) )
26	6, 25	wrdfd 30616	. . 3 |- ( ph -> <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> : ( 0..^ 3 ) --> ( CC ^m NN ) )
27	1, 3, 26	circlemeth 30718	. 2 |- ( ph -> sum_ n e. ( NN (repr ` 3 ) N ) prod_ a e. ( 0..^ 3 ) ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = S. ( 0 (,) 1 ) ( prod_ a e. ( 0..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u N x. x ) ) ) ) _d x )
28		fveq2 6191	. . . . . 6 |- ( a = 0 -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) )
29		fveq2 6191	. . . . . 6 |- ( a = 0 -> ( n ` a ) = ( n ` 0 ) )
30	28, 29	fveq12d 6197	. . . . 5 |- ( a = 0 -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) ` ( n ` 0 ) ) )
31		fveq2 6191	. . . . . 6 |- ( a = 1 -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) )
32		fveq2 6191	. . . . . 6 |- ( a = 1 -> ( n ` a ) = ( n ` 1 ) )
33	31, 32	fveq12d 6197	. . . . 5 |- ( a = 1 -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) ` ( n ` 1 ) ) )
34		fveq2 6191	. . . . . 6 |- ( a = 2 -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) )
35		fveq2 6191	. . . . . 6 |- ( a = 2 -> ( n ` a ) = ( n ` 2 ) )
36	34, 35	fveq12d 6197	. . . . 5 |- ( a = 2 -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) ` ( n ` 2 ) ) )
37	26	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> : ( 0..^ 3 ) --> ( CC ^m NN ) )
38	37	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) /\ a e. ( 0..^ 3 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) e. ( CC ^m NN ) )
39		elmapi 7879	. . . . . . 7 |- ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) e. ( CC ^m NN ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) : NN --> CC )
40	38, 39	syl 17	. . . . . 6 |- ( ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) /\ a e. ( 0..^ 3 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) : NN --> CC )
41		ssid 3624	. . . . . . . . 9 |- NN C_ NN
42	41	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> NN C_ NN )
43	1	nn0zd 11480	. . . . . . . . 9 |- ( ph -> N e. ZZ )
44	43	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> N e. ZZ )
45		3nn0 11310	. . . . . . . . 9 |- 3 e. NN0
46	45	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 3 e. NN0 )
47		simpr 477	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> n e. ( NN (repr ` 3 ) N ) )
48	42, 44, 46, 47	reprf 30690	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> n : ( 0..^ 3 ) --> NN )
49	48	ffvelrnda 6359	. . . . . 6 |- ( ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) /\ a e. ( 0..^ 3 ) ) -> ( n ` a ) e. NN )
50	40, 49	ffvelrnd 6360	. . . . 5 |- ( ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) /\ a e. ( 0..^ 3 ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) e. CC )
51	30, 33, 36, 50	prodfzo03 30681	. . . 4 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> prod_ a e. ( 0..^ 3 ) ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) ` ( n ` 0 ) ) x. ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) ` ( n ` 1 ) ) x. ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) ` ( n ` 2 ) ) ) ) )
52		ovex 6678	. . . . . . . 8 |- (Λ oF x. H ) e. _V
53		s3fv0 13636	. . . . . . . 8 |- ( (Λ oF x. H ) e. _V -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) = (Λ oF x. H ) )
54	52, 53	mp1i 13	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) = (Λ oF x. H ) )
55	54	fveq1d 6193	. . . . . 6 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) ` ( n ` 0 ) ) = ( (Λ oF x. H ) ` ( n ` 0 ) ) )
56		simpl 473	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ph )
57		c0ex 10034	. . . . . . . . . . 11 |- 0 e. _V
58	57	tpid1 4303	. . . . . . . . . 10 |- 0 e. { 0 , 1 , 2 }
59		fzo0to3tp 12554	. . . . . . . . . 10 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
60	58, 59	eleqtrri 2700	. . . . . . . . 9 |- 0 e. ( 0..^ 3 )
61	60	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 0 e. ( 0..^ 3 ) )
62	48, 61	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( n ` 0 ) e. NN )
63		ffn 6045	. . . . . . . . . 10 |- (Λ : NN --> RR -> Λ Fn NN )
64	11, 63	ax-mp 5	. . . . . . . . 9 |- Λ Fn NN
65	64	a1i 11	. . . . . . . 8 |- ( ph -> Λ Fn NN )
66	13	ffnd 6046	. . . . . . . 8 |- ( ph -> H Fn NN )
67		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ ( n ` 0 ) e. NN ) -> (Λ ` ( n ` 0 ) ) = (Λ ` ( n ` 0 ) ) )
68		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ ( n ` 0 ) e. NN ) -> ( H ` ( n ` 0 ) ) = ( H ` ( n ` 0 ) ) )
69	65, 66, 15, 15, 16, 67, 68	ofval 6906	. . . . . . 7 |- ( ( ph /\ ( n ` 0 ) e. NN ) -> ( (Λ oF x. H ) ` ( n ` 0 ) ) = ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) )
70	56, 62, 69	syl2anc 693	. . . . . 6 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( (Λ oF x. H ) ` ( n ` 0 ) ) = ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) )
71	55, 70	eqtrd 2656	. . . . 5 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) ` ( n ` 0 ) ) = ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) )
72		ovex 6678	. . . . . . . . 9 |- (Λ oF x. K ) e. _V
73		s3fv1 13637	. . . . . . . . 9 |- ( (Λ oF x. K ) e. _V -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) = (Λ oF x. K ) )
74	72, 73	mp1i 13	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) = (Λ oF x. K ) )
75	74	fveq1d 6193	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) ` ( n ` 1 ) ) = ( (Λ oF x. K ) ` ( n ` 1 ) ) )
76		1ex 10035	. . . . . . . . . . . 12 |- 1 e. _V
77	76	tpid2 4304	. . . . . . . . . . 11 |- 1 e. { 0 , 1 , 2 }
78	77, 59	eleqtrri 2700	. . . . . . . . . 10 |- 1 e. ( 0..^ 3 )
79	78	a1i 11	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 1 e. ( 0..^ 3 ) )
80	48, 79	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( n ` 1 ) e. NN )
81	21	ffnd 6046	. . . . . . . . 9 |- ( ph -> K Fn NN )
82		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ ( n ` 1 ) e. NN ) -> (Λ ` ( n ` 1 ) ) = (Λ ` ( n ` 1 ) ) )
83		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ ( n ` 1 ) e. NN ) -> ( K ` ( n ` 1 ) ) = ( K ` ( n ` 1 ) ) )
84	65, 81, 15, 15, 16, 82, 83	ofval 6906	. . . . . . . 8 |- ( ( ph /\ ( n ` 1 ) e. NN ) -> ( (Λ oF x. K ) ` ( n ` 1 ) ) = ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) )
85	56, 80, 84	syl2anc 693	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( (Λ oF x. K ) ` ( n ` 1 ) ) = ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) )
86	75, 85	eqtrd 2656	. . . . . 6 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) ` ( n ` 1 ) ) = ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) )
87		s3fv2 13638	. . . . . . . . 9 |- ( (Λ oF x. K ) e. _V -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) = (Λ oF x. K ) )
88	72, 87	mp1i 13	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) = (Λ oF x. K ) )
89	88	fveq1d 6193	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) ` ( n ` 2 ) ) = ( (Λ oF x. K ) ` ( n ` 2 ) ) )
90		2ex 11092	. . . . . . . . . . . 12 |- 2 e. _V
91	90	tpid3 4307	. . . . . . . . . . 11 |- 2 e. { 0 , 1 , 2 }
92	91, 59	eleqtrri 2700	. . . . . . . . . 10 |- 2 e. ( 0..^ 3 )
93	92	a1i 11	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> 2 e. ( 0..^ 3 ) )
94	48, 93	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( n ` 2 ) e. NN )
95		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ ( n ` 2 ) e. NN ) -> (Λ ` ( n ` 2 ) ) = (Λ ` ( n ` 2 ) ) )
96		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ ( n ` 2 ) e. NN ) -> ( K ` ( n ` 2 ) ) = ( K ` ( n ` 2 ) ) )
97	65, 81, 15, 15, 16, 95, 96	ofval 6906	. . . . . . . 8 |- ( ( ph /\ ( n ` 2 ) e. NN ) -> ( (Λ oF x. K ) ` ( n ` 2 ) ) = ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) )
98	56, 94, 97	syl2anc 693	. . . . . . 7 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( (Λ oF x. K ) ` ( n ` 2 ) ) = ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) )
99	89, 98	eqtrd 2656	. . . . . 6 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) ` ( n ` 2 ) ) = ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) )
100	86, 99	oveq12d 6668	. . . . 5 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) ` ( n ` 1 ) ) x. ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) ` ( n ` 2 ) ) ) = ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) )
101	71, 100	oveq12d 6668	. . . 4 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) ` ( n ` 0 ) ) x. ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) ` ( n ` 1 ) ) x. ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) ` ( n ` 2 ) ) ) ) = ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )
102	51, 101	eqtrd 2656	. . 3 |- ( ( ph /\ n e. ( NN (repr ` 3 ) N ) ) -> prod_ a e. ( 0..^ 3 ) ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = ( ( (Λ ` ( n ` 0 ) ) x. ( H ` ( n ` 0 ) ) ) x. ( ( (Λ ` ( n ` 1 ) ) x. ( K ` ( n ` 1 ) ) ) x. ( (Λ ` ( n ` 2 ) ) x. ( K ` ( n ` 2 ) ) ) ) ) )
103	102	sumeq2dv 14433	. 2 |- ( ph -> sum_ n e. ( NN (repr ` 3 ) N ) prod_ a e. ( 0..^ 3 ) ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) ` ( n ` a ) ) = 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 ) ) ) ) ) )
104		nfv 1843	. . . . . 6 |- F/ a ( ph /\ x e. ( 0 (,) 1 ) )
105		nfcv 2764	. . . . . 6 |- F/_ a ( ( (Λ oF x. H )vts N ) ` x )
106		fzofi 12773	. . . . . . 7 |- ( 1..^ 3 ) e. Fin
107	106	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( 1..^ 3 ) e. Fin )
108	57	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> 0 e. _V )
109		eqid 2622	. . . . . . . . 9 |- 0 = 0
110	109	orci 405	. . . . . . . 8 |- ( 0 = 0 \/ 0 = 3 )
111		0elfz 12436	. . . . . . . . 9 |- ( 3 e. NN0 -> 0 e. ( 0 ... 3 ) )
112		elfznelfzob 12574	. . . . . . . . 9 |- ( 0 e. ( 0 ... 3 ) -> ( -. 0 e. ( 1..^ 3 ) <-> ( 0 = 0 \/ 0 = 3 ) ) )
113	45, 111, 112	mp2b 10	. . . . . . . 8 |- ( -. 0 e. ( 1..^ 3 ) <-> ( 0 = 0 \/ 0 = 3 ) )
114	110, 113	mpbir 221	. . . . . . 7 |- -. 0 e. ( 1..^ 3 )
115	114	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> -. 0 e. ( 1..^ 3 ) )
116	1	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> N e. NN0 )
117		ioossre 12235	. . . . . . . . . . 11 |- ( 0 (,) 1 ) C_ RR
118		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
119	117, 118	sstri 3612	. . . . . . . . . 10 |- ( 0 (,) 1 ) C_ CC
120	119	a1i 11	. . . . . . . . 9 |- ( ph -> ( 0 (,) 1 ) C_ CC )
121	120	sselda 3603	. . . . . . . 8 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> x e. CC )
122	121	adantr 481	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> x e. CC )
123	26	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> : ( 0..^ 3 ) --> ( CC ^m NN ) )
124		fzo0ss1 12498	. . . . . . . . . . 11 |- ( 1..^ 3 ) C_ ( 0..^ 3 )
125	124	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( 1..^ 3 ) C_ ( 0..^ 3 ) )
126	125	sselda 3603	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> a e. ( 0..^ 3 ) )
127	123, 126	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) e. ( CC ^m NN ) )
128	127, 39	syl 17	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) : NN --> CC )
129	116, 122, 128	vtscl 30716	. . . . . 6 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) e. CC )
130	52, 53	ax-mp 5	. . . . . . . . 9 |- ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 0 ) = (Λ oF x. H )
131	28, 130	syl6eq 2672	. . . . . . . 8 |- ( a = 0 -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = (Λ oF x. H ) )
132	131	oveq1d 6665	. . . . . . 7 |- ( a = 0 -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) = ( (Λ oF x. H )vts N ) )
133	132	fveq1d 6193	. . . . . 6 |- ( a = 0 -> ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = ( ( (Λ oF x. H )vts N ) ` x ) )
134	1	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> N e. NN0 )
135	17	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> (Λ oF x. H ) : NN --> CC )
136	134, 121, 135	vtscl 30716	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( ( (Λ oF x. H )vts N ) ` x ) e. CC )
137	104, 105, 107, 108, 115, 129, 133, 136	fprodsplitsn 14720	. . . . 5 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> prod_ a e. ( ( 1..^ 3 ) u. { 0 } ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = ( prod_ a e. ( 1..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) x. ( ( (Λ oF x. H )vts N ) ` x ) ) )
138		uncom 3757	. . . . . . . 8 |- ( ( 1..^ 3 ) u. { 0 } ) = ( { 0 } u. ( 1..^ 3 ) )
139		fzo0sn0fzo1 12557	. . . . . . . . 9 |- ( 3 e. NN -> ( 0..^ 3 ) = ( { 0 } u. ( 1..^ 3 ) ) )
140	2, 139	ax-mp 5	. . . . . . . 8 |- ( 0..^ 3 ) = ( { 0 } u. ( 1..^ 3 ) )
141	138, 140	eqtr4i 2647	. . . . . . 7 |- ( ( 1..^ 3 ) u. { 0 } ) = ( 0..^ 3 )
142	141	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( ( 1..^ 3 ) u. { 0 } ) = ( 0..^ 3 ) )
143	142	prodeq1d 14651	. . . . 5 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> prod_ a e. ( ( 1..^ 3 ) u. { 0 } ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = prod_ a e. ( 0..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) )
144		fzo13pr 12552	. . . . . . . . . . . . . . 15 |- ( 1..^ 3 ) = { 1 , 2 }
145	144	eleq2i 2693	. . . . . . . . . . . . . 14 |- ( a e. ( 1..^ 3 ) <-> a e. { 1 , 2 } )
146		vex 3203	. . . . . . . . . . . . . . 15 |- a e. _V
147	146	elpr 4198	. . . . . . . . . . . . . 14 |- ( a e. { 1 , 2 } <-> ( a = 1 \/ a = 2 ) )
148	145, 147	bitri 264	. . . . . . . . . . . . 13 |- ( a e. ( 1..^ 3 ) <-> ( a = 1 \/ a = 2 ) )
149	31	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a = 1 ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) )
150	72, 73	mp1i 13	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a = 1 ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 1 ) = (Λ oF x. K ) )
151	149, 150	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ph /\ a = 1 ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = (Λ oF x. K ) )
152	34	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a = 2 ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) )
153	72, 87	mp1i 13	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a = 2 ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` 2 ) = (Λ oF x. K ) )
154	152, 153	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ph /\ a = 2 ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = (Λ oF x. K ) )
155	151, 154	jaodan 826	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a = 1 \/ a = 2 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = (Λ oF x. K ) )
156	148, 155	sylan2b 492	. . . . . . . . . . . 12 |- ( ( ph /\ a e. ( 1..^ 3 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = (Λ oF x. K ) )
157	156	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a ) = (Λ oF x. K ) )
158	157	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) = ( (Λ oF x. K )vts N ) )
159	158	fveq1d 6193	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0 (,) 1 ) ) /\ a e. ( 1..^ 3 ) ) -> ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = ( ( (Λ oF x. K )vts N ) ` x ) )
160	159	prodeq2dv 14653	. . . . . . . 8 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> prod_ a e. ( 1..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = prod_ a e. ( 1..^ 3 ) ( ( (Λ oF x. K )vts N ) ` x ) )
161	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> (Λ oF x. K ) : NN --> CC )
162	134, 121, 161	vtscl 30716	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( ( (Λ oF x. K )vts N ) ` x ) e. CC )
163		fprodconst 14708	. . . . . . . . 9 |- ( ( ( 1..^ 3 ) e. Fin /\ ( ( (Λ oF x. K )vts N ) ` x ) e. CC ) -> prod_ a e. ( 1..^ 3 ) ( ( (Λ oF x. K )vts N ) ` x ) = ( ( ( (Λ oF x. K )vts N ) ` x ) ^ ( # ` ( 1..^ 3 ) ) ) )
164	107, 162, 163	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> prod_ a e. ( 1..^ 3 ) ( ( (Λ oF x. K )vts N ) ` x ) = ( ( ( (Λ oF x. K )vts N ) ` x ) ^ ( # ` ( 1..^ 3 ) ) ) )
165		nnuz 11723	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
166	2, 165	eleqtri 2699	. . . . . . . . . . . 12 |- 3 e. ( ZZ>= ` 1 )
167		hashfzo 13216	. . . . . . . . . . . 12 |- ( 3 e. ( ZZ>= ` 1 ) -> ( # ` ( 1..^ 3 ) ) = ( 3 - 1 ) )
168	166, 167	ax-mp 5	. . . . . . . . . . 11 |- ( # ` ( 1..^ 3 ) ) = ( 3 - 1 )
169		3m1e2 11137	. . . . . . . . . . 11 |- ( 3 - 1 ) = 2
170	168, 169	eqtri 2644	. . . . . . . . . 10 |- ( # ` ( 1..^ 3 ) ) = 2
171	170	a1i 11	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( # ` ( 1..^ 3 ) ) = 2 )
172	171	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( ( ( (Λ oF x. K )vts N ) ` x ) ^ ( # ` ( 1..^ 3 ) ) ) = ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) )
173	160, 164, 172	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> prod_ a e. ( 1..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) )
174	173	oveq1d 6665	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( prod_ a e. ( 1..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) x. ( ( (Λ oF x. H )vts N ) ` x ) ) = ( ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) x. ( ( (Λ oF x. H )vts N ) ` x ) ) )
175	162	sqcld 13006	. . . . . . 7 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) e. CC )
176	136, 175	mulcomd 10061	. . . . . 6 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( ( ( (Λ oF x. H )vts N ) ` x ) x. ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) ) = ( ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) x. ( ( (Λ oF x. H )vts N ) ` x ) ) )
177	174, 176	eqtr4d 2659	. . . . 5 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( prod_ a e. ( 1..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) x. ( ( (Λ oF x. H )vts N ) ` x ) ) = ( ( ( (Λ oF x. H )vts N ) ` x ) x. ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) ) )
178	137, 143, 177	3eqtr3d 2664	. . . 4 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> prod_ a e. ( 0..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) = ( ( ( (Λ oF x. H )vts N ) ` x ) x. ( ( ( (Λ oF x. K )vts N ) ` x ) ^ 2 ) ) )
179	178	oveq1d 6665	. . 3 |- ( ( ph /\ x e. ( 0 (,) 1 ) ) -> ( prod_ a e. ( 0..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u N x. x ) ) ) ) = ( ( ( ( (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 ) ) ) ) )
180	179	itgeq2dv 23548	. 2 |- ( ph -> S. ( 0 (,) 1 ) ( prod_ a e. ( 0..^ 3 ) ( ( ( <" (Λ oF x. H ) (Λ oF x. K ) (Λ oF x. K ) "> ` a )vts N ) ` x ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( -u N x. x ) ) ) ) _d x = 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 )
181	27, 103, 180	3eqtr3d 2664	1 |- ( 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 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   {ctp 4181   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   x. cmul 9941   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   (,)cioo 12175   ...cfz 12326  ..^cfzo 12465   ^cexp 12860   #chash 13117   <"cs3 13587   sum_csu 14416   prod_cprod 14635   expce 14792   picpi 14797   S.citg 23387  Λcvma 24818  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-inf2 8538  ax-cc 9257  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

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-card 8765  df-acn 8768  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-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-sin 14800  df-cos 14801  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-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-log 24303  df-vma 24824  df-repr 30687  df-vts 30714

This theorem is referenced by:  tgoldbachgtde  30738