Theorem fsumvma 24938

Description: Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)

Hypotheses

Ref	Expression
fsumvma.1	|- ( x = ( p ^ k ) -> B = C )
fsumvma.2	|- ( ph -> A e. Fin )
fsumvma.3	|- ( ph -> A C_ NN )
fsumvma.4	|- ( ph -> P e. Fin )
fsumvma.5	|- ( ph -> ( ( p e. P /\ k e. K ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) ) )
fsumvma.6	|- ( ( ph /\ x e. A ) -> B e. CC )
fsumvma.7	|- ( ( ph /\ ( x e. A /\ (Λ ` x ) = 0 ) ) -> B = 0 )

Assertion

Ref	Expression
fsumvma	|- ( ph -> sum_ x e. A B = sum_ p e. P sum_ k e. K C )

Distinct variable groups:   k, p, x, A   x, C   k, K, x   ph, k, p, x   B, k, p   P, k, p, x

Allowed substitution hints:   B( x)   C( k, p)   K( p)

Proof of Theorem fsumvma

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

Step	Hyp	Ref	Expression
1		fvexd 6203	. . . 4 |- ( z = <. p , k >. -> ( ^ ` z ) e. _V )
2		fveq2 6191	. . . . . . . 8 |- ( z = <. p , k >. -> ( ^ ` z ) = ( ^ ` <. p , k >. ) )
3		df-ov 6653	. . . . . . . 8 |- ( p ^ k ) = ( ^ ` <. p , k >. )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( z = <. p , k >. -> ( ^ ` z ) = ( p ^ k ) )
5	4	eqeq2d 2632	. . . . . 6 |- ( z = <. p , k >. -> ( x = ( ^ ` z ) <-> x = ( p ^ k ) ) )
6	5	biimpa 501	. . . . 5 |- ( ( z = <. p , k >. /\ x = ( ^ ` z ) ) -> x = ( p ^ k ) )
7		fsumvma.1	. . . . 5 |- ( x = ( p ^ k ) -> B = C )
8	6, 7	syl 17	. . . 4 |- ( ( z = <. p , k >. /\ x = ( ^ ` z ) ) -> B = C )
9	1, 8	csbied 3560	. . 3 |- ( z = <. p , k >. -> [_ ( ^ ` z ) / x ]_ B = C )
10		fsumvma.4	. . 3 |- ( ph -> P e. Fin )
11		fsumvma.2	. . . . 5 |- ( ph -> A e. Fin )
12	11	adantr 481	. . . 4 |- ( ( ph /\ p e. P ) -> A e. Fin )
13		fsumvma.5	. . . . . . . . 9 |- ( ph -> ( ( p e. P /\ k e. K ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) ) )
14	13	biimpd 219	. . . . . . . 8 |- ( ph -> ( ( p e. P /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) ) )
15	14	impl 650	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) )
16	15	simprd 479	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p ^ k ) e. A )
17	16	ex 450	. . . . 5 |- ( ( ph /\ p e. P ) -> ( k e. K -> ( p ^ k ) e. A ) )
18	15	simpld 475	. . . . . . . . 9 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p e. Prime /\ k e. NN ) )
19	18	simpld 475	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> p e. Prime )
20	19	adantrr 753	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> p e. Prime )
21	18	simprd 479	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> k e. NN )
22	21	adantrr 753	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> k e. NN )
23	21	ex 450	. . . . . . . . . 10 |- ( ( ph /\ p e. P ) -> ( k e. K -> k e. NN ) )
24	23	ssrdv 3609	. . . . . . . . 9 |- ( ( ph /\ p e. P ) -> K C_ NN )
25	24	sselda 3603	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ z e. K ) -> z e. NN )
26	25	adantrl 752	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> z e. NN )
27		eqid 2622	. . . . . . . 8 |- p = p
28		prmexpb 15430	. . . . . . . . 9 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> ( p = p /\ k = z ) ) )
29	28	baibd 948	. . . . . . . 8 |- ( ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) /\ p = p ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
30	27, 29	mpan2 707	. . . . . . 7 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
31	20, 20, 22, 26, 30	syl22anc 1327	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
32	31	ex 450	. . . . 5 |- ( ( ph /\ p e. P ) -> ( ( k e. K /\ z e. K ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) ) )
33	17, 32	dom2lem 7995	. . . 4 |- ( ( ph /\ p e. P ) -> ( k e. K |-> ( p ^ k ) ) : K -1-1-> A )
34		f1fi 8253	. . . 4 |- ( ( A e. Fin /\ ( k e. K |-> ( p ^ k ) ) : K -1-1-> A ) -> K e. Fin )
35	12, 33, 34	syl2anc 693	. . 3 |- ( ( ph /\ p e. P ) -> K e. Fin )
36	13	simplbda 654	. . . 4 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( p ^ k ) e. A )
37		fsumvma.6	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. CC )
38	37	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. A B e. CC )
39	38	adantr 481	. . . 4 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> A. x e. A B e. CC )
40	7	eleq1d 2686	. . . . 5 |- ( x = ( p ^ k ) -> ( B e. CC <-> C e. CC ) )
41	40	rspcv 3305	. . . 4 |- ( ( p ^ k ) e. A -> ( A. x e. A B e. CC -> C e. CC ) )
42	36, 39, 41	sylc 65	. . 3 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> C e. CC )
43	9, 10, 35, 42	fsum2d 14502	. 2 |- ( ph -> sum_ p e. P sum_ k e. K C = sum_ z e. U_ p e. P ( { p } X. K ) [_ ( ^ ` z ) / x ]_ B )
44		nfcv 2764	. . . 4 |- F/_ y B
45		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ B
46		csbeq1a 3542	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
47	44, 45, 46	cbvsumi 14427	. . 3 |- sum_ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) B = sum_ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) [_ y / x ]_ B
48		csbeq1 3536	. . . 4 |- ( y = ( ^ ` z ) -> [_ y / x ]_ B = [_ ( ^ ` z ) / x ]_ B )
49		snfi 8038	. . . . . . 7 |- { p } e. Fin
50		xpfi 8231	. . . . . . 7 |- ( ( { p } e. Fin /\ K e. Fin ) -> ( { p } X. K ) e. Fin )
51	49, 35, 50	sylancr 695	. . . . . 6 |- ( ( ph /\ p e. P ) -> ( { p } X. K ) e. Fin )
52	51	ralrimiva 2966	. . . . 5 |- ( ph -> A. p e. P ( { p } X. K ) e. Fin )
53		iunfi 8254	. . . . 5 |- ( ( P e. Fin /\ A. p e. P ( { p } X. K ) e. Fin ) -> U_ p e. P ( { p } X. K ) e. Fin )
54	10, 52, 53	syl2anc 693	. . . 4 |- ( ph -> U_ p e. P ( { p } X. K ) e. Fin )
55		fvex 6201	. . . . . . 7 |- ( ^ ` a ) e. _V
56	55	2a1i 12	. . . . . 6 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. _V ) )
57		eliunxp 5259	. . . . . . . . 9 |- ( a e. U_ p e. P ( { p } X. K ) <-> E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) )
58	13	simprbda 653	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( p e. Prime /\ k e. NN ) )
59		opelxp 5146	. . . . . . . . . . . . . 14 |- ( <. p , k >. e. ( Prime X. NN ) <-> ( p e. Prime /\ k e. NN ) )
60	58, 59	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> <. p , k >. e. ( Prime X. NN ) )
61		eleq1 2689	. . . . . . . . . . . . 13 |- ( a = <. p , k >. -> ( a e. ( Prime X. NN ) <-> <. p , k >. e. ( Prime X. NN ) ) )
62	60, 61	syl5ibrcom 237	. . . . . . . . . . . 12 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( a = <. p , k >. -> a e. ( Prime X. NN ) ) )
63	62	impancom 456	. . . . . . . . . . 11 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> a e. ( Prime X. NN ) ) )
64	63	expimpd 629	. . . . . . . . . 10 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
65	64	exlimdvv 1862	. . . . . . . . 9 |- ( ph -> ( E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
66	57, 65	syl5bi 232	. . . . . . . 8 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> a e. ( Prime X. NN ) ) )
67	66	ssrdv 3609	. . . . . . . . 9 |- ( ph -> U_ p e. P ( { p } X. K ) C_ ( Prime X. NN ) )
68	67	sseld 3602	. . . . . . . 8 |- ( ph -> ( b e. U_ p e. P ( { p } X. K ) -> b e. ( Prime X. NN ) ) )
69	66, 68	anim12d 586	. . . . . . 7 |- ( ph -> ( ( a e. U_ p e. P ( { p } X. K ) /\ b e. U_ p e. P ( { p } X. K ) ) -> ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) ) )
70		1st2nd2 7205	. . . . . . . . . . 11 |- ( a e. ( Prime X. NN ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
71	70	fveq2d 6195	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
72		df-ov 6653	. . . . . . . . . 10 |- ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
73	71, 72	syl6eqr 2674	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ( 1st ` a ) ^ ( 2nd ` a ) ) )
74		1st2nd2 7205	. . . . . . . . . . 11 |- ( b e. ( Prime X. NN ) -> b = <. ( 1st ` b ) , ( 2nd ` b ) >. )
75	74	fveq2d 6195	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. ) )
76		df-ov 6653	. . . . . . . . . 10 |- ( ( 1st ` b ) ^ ( 2nd ` b ) ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. )
77	75, 76	syl6eqr 2674	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) )
78	73, 77	eqeqan12d 2638	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) ) )
79		xp1st 7198	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 1st ` a ) e. Prime )
80		xp2nd 7199	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 2nd ` a ) e. NN )
81	79, 80	jca 554	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ( 1st ` a ) e. Prime /\ ( 2nd ` a ) e. NN ) )
82		xp1st 7198	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 1st ` b ) e. Prime )
83		xp2nd 7199	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 2nd ` b ) e. NN )
84	82, 83	jca 554	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ( 1st ` b ) e. Prime /\ ( 2nd ` b ) e. NN ) )
85		prmexpb 15430	. . . . . . . . . 10 |- ( ( ( ( 1st ` a ) e. Prime /\ ( 1st ` b ) e. Prime ) /\ ( ( 2nd ` a ) e. NN /\ ( 2nd ` b ) e. NN ) ) -> ( ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) <-> ( ( 1st ` a ) = ( 1st ` b ) /\ ( 2nd ` a ) = ( 2nd ` b ) ) ) )
86	85	an4s 869	. . . . . . . . 9 |- ( ( ( ( 1st ` a ) e. Prime /\ ( 2nd ` a ) e. NN ) /\ ( ( 1st ` b ) e. Prime /\ ( 2nd ` b ) e. NN ) ) -> ( ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) <-> ( ( 1st ` a ) = ( 1st ` b ) /\ ( 2nd ` a ) = ( 2nd ` b ) ) ) )
87	81, 84, 86	syl2an 494	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) <-> ( ( 1st ` a ) = ( 1st ` b ) /\ ( 2nd ` a ) = ( 2nd ` b ) ) ) )
88		xpopth 7207	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ( 1st ` a ) = ( 1st ` b ) /\ ( 2nd ` a ) = ( 2nd ` b ) ) <-> a = b ) )
89	78, 87, 88	3bitrd 294	. . . . . . 7 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> a = b ) )
90	69, 89	syl6 35	. . . . . 6 |- ( ph -> ( ( a e. U_ p e. P ( { p } X. K ) /\ b e. U_ p e. P ( { p } X. K ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> a = b ) ) )
91	56, 90	dom2lem 7995	. . . . 5 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-> _V )
92		f1f1orn 6148	. . . . 5 |- ( ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-> _V -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-onto-> ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) )
93	91, 92	syl 17	. . . 4 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-onto-> ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) )
94		fveq2 6191	. . . . . 6 |- ( a = z -> ( ^ ` a ) = ( ^ ` z ) )
95		eqid 2622	. . . . . 6 |- ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) = ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) )
96		fvex 6201	. . . . . 6 |- ( ^ ` z ) e. _V
97	94, 95, 96	fvmpt 6282	. . . . 5 |- ( z e. U_ p e. P ( { p } X. K ) -> ( ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ` z ) = ( ^ ` z ) )
98	97	adantl 482	. . . 4 |- ( ( ph /\ z e. U_ p e. P ( { p } X. K ) ) -> ( ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ` z ) = ( ^ ` z ) )
99		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( a = <. p , k >. -> ( ^ ` a ) = ( ^ ` <. p , k >. ) )
100	99, 3	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( a = <. p , k >. -> ( ^ ` a ) = ( p ^ k ) )
101	100	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( a = <. p , k >. -> ( ( ^ ` a ) e. A <-> ( p ^ k ) e. A ) )
102	36, 101	syl5ibrcom 237	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( a = <. p , k >. -> ( ^ ` a ) e. A ) )
103	102	impancom 456	. . . . . . . . . . . 12 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> ( ^ ` a ) e. A ) )
104	103	expimpd 629	. . . . . . . . . . 11 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
105	104	exlimdvv 1862	. . . . . . . . . 10 |- ( ph -> ( E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
106	57, 105	syl5bi 232	. . . . . . . . 9 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. A ) )
107	106	imp 445	. . . . . . . 8 |- ( ( ph /\ a e. U_ p e. P ( { p } X. K ) ) -> ( ^ ` a ) e. A )
108	107, 95	fmptd 6385	. . . . . . 7 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) --> A )
109		frn 6053	. . . . . . 7 |- ( ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) --> A -> ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) C_ A )
110	108, 109	syl 17	. . . . . 6 |- ( ph -> ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) C_ A )
111	110	sselda 3603	. . . . 5 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> y e. A )
112	45	nfel1 2779	. . . . . . 7 |- F/ x [_ y / x ]_ B e. CC
113	46	eleq1d 2686	. . . . . . 7 |- ( x = y -> ( B e. CC <-> [_ y / x ]_ B e. CC ) )
114	112, 113	rspc 3303	. . . . . 6 |- ( y e. A -> ( A. x e. A B e. CC -> [_ y / x ]_ B e. CC ) )
115	38, 114	mpan9 486	. . . . 5 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. CC )
116	111, 115	syldan 487	. . . 4 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> [_ y / x ]_ B e. CC )
117	48, 54, 93, 98, 116	fsumf1o 14454	. . 3 |- ( ph -> sum_ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) [_ y / x ]_ B = sum_ z e. U_ p e. P ( { p } X. K ) [_ ( ^ ` z ) / x ]_ B )
118	47, 117	syl5eq 2668	. 2 |- ( ph -> sum_ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) B = sum_ z e. U_ p e. P ( { p } X. K ) [_ ( ^ ` z ) / x ]_ B )
119	110	sselda 3603	. . . 4 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> x e. A )
120	119, 37	syldan 487	. . 3 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B e. CC )
121		eldif 3584	. . . . 5 |- ( x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) <-> ( x e. A /\ -. x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) )
122	95, 55	elrnmpti 5376	. . . . . . . . . 10 |- ( x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) <-> E. a e. U_ p e. P ( { p } X. K ) x = ( ^ ` a ) )
123	100	eqeq2d 2632	. . . . . . . . . . 11 |- ( a = <. p , k >. -> ( x = ( ^ ` a ) <-> x = ( p ^ k ) ) )
124	123	rexiunxp 5262	. . . . . . . . . 10 |- ( E. a e. U_ p e. P ( { p } X. K ) x = ( ^ ` a ) <-> E. p e. P E. k e. K x = ( p ^ k ) )
125	122, 124	bitri 264	. . . . . . . . 9 |- ( x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) <-> E. p e. P E. k e. K x = ( p ^ k ) )
126		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> x = ( p ^ k ) )
127		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> x e. A )
128	126, 127	eqeltrrd 2702	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( p ^ k ) e. A )
129	13	rbaibd 949	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( p ^ k ) e. A ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
130	129	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ ( p ^ k ) e. A ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
131	128, 130	syldan 487	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
132	131	pm5.32da 673	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> ( ( x = ( p ^ k ) /\ ( p e. P /\ k e. K ) ) <-> ( x = ( p ^ k ) /\ ( p e. Prime /\ k e. NN ) ) ) )
133		ancom 466	. . . . . . . . . . . . 13 |- ( ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. P /\ k e. K ) ) )
134		ancom 466	. . . . . . . . . . . . 13 |- ( ( ( p e. Prime /\ k e. NN ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. Prime /\ k e. NN ) ) )
135	132, 133, 134	3bitr4g 303	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ( ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ x = ( p ^ k ) ) ) )
136	135	2exbidv 1852	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( E. p E. k ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) <-> E. p E. k ( ( p e. Prime /\ k e. NN ) /\ x = ( p ^ k ) ) ) )
137		r2ex 3061	. . . . . . . . . . 11 |- ( E. p e. P E. k e. K x = ( p ^ k ) <-> E. p E. k ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) )
138		r2ex 3061	. . . . . . . . . . 11 |- ( E. p e. Prime E. k e. NN x = ( p ^ k ) <-> E. p E. k ( ( p e. Prime /\ k e. NN ) /\ x = ( p ^ k ) ) )
139	136, 137, 138	3bitr4g 303	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( E. p e. P E. k e. K x = ( p ^ k ) <-> E. p e. Prime E. k e. NN x = ( p ^ k ) ) )
140		fsumvma.3	. . . . . . . . . . . 12 |- ( ph -> A C_ NN )
141	140	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. NN )
142		isppw2 24841	. . . . . . . . . . 11 |- ( x e. NN -> ( (Λ ` x ) =/= 0 <-> E. p e. Prime E. k e. NN x = ( p ^ k ) ) )
143	141, 142	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( (Λ ` x ) =/= 0 <-> E. p e. Prime E. k e. NN x = ( p ^ k ) ) )
144	139, 143	bitr4d 271	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( E. p e. P E. k e. K x = ( p ^ k ) <-> (Λ ` x ) =/= 0 ) )
145	125, 144	syl5bb 272	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) <-> (Λ ` x ) =/= 0 ) )
146	145	necon2bbid 2837	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( (Λ ` x ) = 0 <-> -. x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) )
147	146	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( x e. A /\ (Λ ` x ) = 0 ) <-> ( x e. A /\ -. x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) ) )
148		fsumvma.7	. . . . . . 7 |- ( ( ph /\ ( x e. A /\ (Λ ` x ) = 0 ) ) -> B = 0 )
149	148	ex 450	. . . . . 6 |- ( ph -> ( ( x e. A /\ (Λ ` x ) = 0 ) -> B = 0 ) )
150	147, 149	sylbird 250	. . . . 5 |- ( ph -> ( ( x e. A /\ -. x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B = 0 ) )
151	121, 150	syl5bi 232	. . . 4 |- ( ph -> ( x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B = 0 ) )
152	151	imp 445	. . 3 |- ( ( ph /\ x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) ) -> B = 0 )
153	110, 120, 152, 11	fsumss 14456	. 2 |- ( ph -> sum_ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) B = sum_ x e. A B )
154	43, 118, 153	3eqtr2rd 2663	1 |- ( ph -> sum_ x e. A B = sum_ p e. P sum_ k e. K C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   [_csb 3533   \ cdif 3571   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   CCcc 9934   0cc0 9936   NNcn 11020   ^cexp 12860   sum_csu 14416   Primecprime 15385  Λcvma 24818

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-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-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-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-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-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-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-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-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-limc 23630  df-dv 23631  df-log 24303  df-vma 24824

This theorem is referenced by:  fsumvma2  24939  vmasum  24941