Theorem fsumvma 23869

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		fvex 5859	. . . . 5 |- ( ^ ` z ) e. _V
2	1	a1i 11	. . . 4 |- ( z = <. p , k >. -> ( ^ ` z ) e. _V )
3		fveq2 5849	. . . . . . . 8 |- ( z = <. p , k >. -> ( ^ ` z ) = ( ^ ` <. p , k >. ) )
4		df-ov 6281	. . . . . . . 8 |- ( p ^ k ) = ( ^ ` <. p , k >. )
5	3, 4	syl6eqr 2461	. . . . . . 7 |- ( z = <. p , k >. -> ( ^ ` z ) = ( p ^ k ) )
6	5	eqeq2d 2416	. . . . . 6 |- ( z = <. p , k >. -> ( x = ( ^ ` z ) <-> x = ( p ^ k ) ) )
7	6	biimpa 482	. . . . 5 |- ( ( z = <. p , k >. /\ x = ( ^ ` z ) ) -> x = ( p ^ k ) )
8		fsumvma.1	. . . . 5 |- ( x = ( p ^ k ) -> B = C )
9	7, 8	syl 17	. . . 4 |- ( ( z = <. p , k >. /\ x = ( ^ ` z ) ) -> B = C )
10	2, 9	csbied 3400	. . 3 |- ( z = <. p , k >. -> [_ ( ^ ` z ) / x ]_ B = C )
11		fsumvma.4	. . 3 |- ( ph -> P e. Fin )
12		fsumvma.2	. . . . 5 |- ( ph -> A e. Fin )
13	12	adantr 463	. . . 4 |- ( ( ph /\ p e. P ) -> A e. Fin )
14		fsumvma.5	. . . . . . . . 9 |- ( ph -> ( ( p e. P /\ k e. K ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) ) )
15	14	biimpd 207	. . . . . . . 8 |- ( ph -> ( ( p e. P /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) ) )
16	15	impl 618	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) )
17	16	simprd 461	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p ^ k ) e. A )
18	17	ex 432	. . . . 5 |- ( ( ph /\ p e. P ) -> ( k e. K -> ( p ^ k ) e. A ) )
19	16	simpld 457	. . . . . . . . 9 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p e. Prime /\ k e. NN ) )
20	19	simpld 457	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> p e. Prime )
21	20	adantrr 715	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> p e. Prime )
22	19	simprd 461	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> k e. NN )
23	22	adantrr 715	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> k e. NN )
24	22	ex 432	. . . . . . . . . 10 |- ( ( ph /\ p e. P ) -> ( k e. K -> k e. NN ) )
25	24	ssrdv 3448	. . . . . . . . 9 |- ( ( ph /\ p e. P ) -> K C_ NN )
26	25	sselda 3442	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ z e. K ) -> z e. NN )
27	26	adantrl 714	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> z e. NN )
28		eqid 2402	. . . . . . . 8 |- p = p
29		prmexpb 14467	. . . . . . . . 9 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> ( p = p /\ k = z ) ) )
30	29	baibd 910	. . . . . . . 8 |- ( ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) /\ p = p ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
31	28, 30	mpan2 669	. . . . . . 7 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
32	21, 21, 23, 27, 31	syl22anc 1231	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
33	32	ex 432	. . . . 5 |- ( ( ph /\ p e. P ) -> ( ( k e. K /\ z e. K ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) ) )
34	18, 33	dom2lem 7593	. . . 4 |- ( ( ph /\ p e. P ) -> ( k e. K |-> ( p ^ k ) ) : K -1-1-> A )
35		f1fi 7841	. . . 4 |- ( ( A e. Fin /\ ( k e. K |-> ( p ^ k ) ) : K -1-1-> A ) -> K e. Fin )
36	13, 34, 35	syl2anc 659	. . 3 |- ( ( ph /\ p e. P ) -> K e. Fin )
37	14	simplbda 622	. . . 4 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( p ^ k ) e. A )
38		fsumvma.6	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. CC )
39	38	ralrimiva 2818	. . . . 5 |- ( ph -> A. x e. A B e. CC )
40	39	adantr 463	. . . 4 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> A. x e. A B e. CC )
41	8	eleq1d 2471	. . . . 5 |- ( x = ( p ^ k ) -> ( B e. CC <-> C e. CC ) )
42	41	rspcv 3156	. . . 4 |- ( ( p ^ k ) e. A -> ( A. x e. A B e. CC -> C e. CC ) )
43	37, 40, 42	sylc 59	. . 3 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> C e. CC )
44	10, 11, 36, 43	fsum2d 13737	. 2 |- ( ph -> sum_ p e. P sum_ k e. K C = sum_ z e. U_ p e. P ( { p } X. K ) [_ ( ^ ` z ) / x ]_ B )
45		nfcv 2564	. . . 4 |- F/_ y B
46		nfcsb1v 3389	. . . 4 |- F/_ x [_ y / x ]_ B
47		csbeq1a 3382	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
48	45, 46, 47	cbvsumi 13668	. . 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
49		csbeq1 3376	. . . 4 |- ( y = ( ^ ` z ) -> [_ y / x ]_ B = [_ ( ^ ` z ) / x ]_ B )
50		snfi 7634	. . . . . . 7 |- { p } e. Fin
51		xpfi 7825	. . . . . . 7 |- ( ( { p } e. Fin /\ K e. Fin ) -> ( { p } X. K ) e. Fin )
52	50, 36, 51	sylancr 661	. . . . . 6 |- ( ( ph /\ p e. P ) -> ( { p } X. K ) e. Fin )
53	52	ralrimiva 2818	. . . . 5 |- ( ph -> A. p e. P ( { p } X. K ) e. Fin )
54		iunfi 7842	. . . . 5 |- ( ( P e. Fin /\ A. p e. P ( { p } X. K ) e. Fin ) -> U_ p e. P ( { p } X. K ) e. Fin )
55	11, 53, 54	syl2anc 659	. . . 4 |- ( ph -> U_ p e. P ( { p } X. K ) e. Fin )
56		fvex 5859	. . . . . . 7 |- ( ^ ` a ) e. _V
57	56	a1ii 12	. . . . . 6 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. _V ) )
58		eliunxp 4961	. . . . . . . . 9 |- ( a e. U_ p e. P ( { p } X. K ) <-> E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) )
59	14	simprbda 621	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( p e. Prime /\ k e. NN ) )
60		opelxp 4853	. . . . . . . . . . . . . 14 |- ( <. p , k >. e. ( Prime X. NN ) <-> ( p e. Prime /\ k e. NN ) )
61	59, 60	sylibr 212	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> <. p , k >. e. ( Prime X. NN ) )
62		eleq1 2474	. . . . . . . . . . . . 13 |- ( a = <. p , k >. -> ( a e. ( Prime X. NN ) <-> <. p , k >. e. ( Prime X. NN ) ) )
63	61, 62	syl5ibrcom 222	. . . . . . . . . . . 12 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( a = <. p , k >. -> a e. ( Prime X. NN ) ) )
64	63	impancom 438	. . . . . . . . . . 11 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> a e. ( Prime X. NN ) ) )
65	64	expimpd 601	. . . . . . . . . 10 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
66	65	exlimdvv 1746	. . . . . . . . 9 |- ( ph -> ( E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
67	58, 66	syl5bi 217	. . . . . . . 8 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> a e. ( Prime X. NN ) ) )
68	67	ssrdv 3448	. . . . . . . . 9 |- ( ph -> U_ p e. P ( { p } X. K ) C_ ( Prime X. NN ) )
69	68	sseld 3441	. . . . . . . 8 |- ( ph -> ( b e. U_ p e. P ( { p } X. K ) -> b e. ( Prime X. NN ) ) )
70	67, 69	anim12d 561	. . . . . . 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 ) ) ) )
71		1st2nd2 6821	. . . . . . . . . . 11 |- ( a e. ( Prime X. NN ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
72	71	fveq2d 5853	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
73		df-ov 6281	. . . . . . . . . 10 |- ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
74	72, 73	syl6eqr 2461	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ( 1st ` a ) ^ ( 2nd ` a ) ) )
75		1st2nd2 6821	. . . . . . . . . . 11 |- ( b e. ( Prime X. NN ) -> b = <. ( 1st ` b ) , ( 2nd ` b ) >. )
76	75	fveq2d 5853	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. ) )
77		df-ov 6281	. . . . . . . . . 10 |- ( ( 1st ` b ) ^ ( 2nd ` b ) ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. )
78	76, 77	syl6eqr 2461	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) )
79	74, 78	eqeqan12d 2425	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) ) )
80		xp1st 6814	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 1st ` a ) e. Prime )
81		xp2nd 6815	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 2nd ` a ) e. NN )
82	80, 81	jca 530	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ( 1st ` a ) e. Prime /\ ( 2nd ` a ) e. NN ) )
83		xp1st 6814	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 1st ` b ) e. Prime )
84		xp2nd 6815	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 2nd ` b ) e. NN )
85	83, 84	jca 530	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ( 1st ` b ) e. Prime /\ ( 2nd ` b ) e. NN ) )
86		prmexpb 14467	. . . . . . . . . 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 ) ) ) )
87	86	an4s 827	. . . . . . . . 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 ) ) ) )
88	82, 85, 87	syl2an 475	. . . . . . . 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 ) ) ) )
89		xpopth 6823	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ( 1st ` a ) = ( 1st ` b ) /\ ( 2nd ` a ) = ( 2nd ` b ) ) <-> a = b ) )
90	79, 88, 89	3bitrd 279	. . . . . . 7 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> a = b ) )
91	70, 90	syl6 31	. . . . . 6 |- ( ph -> ( ( a e. U_ p e. P ( { p } X. K ) /\ b e. U_ p e. P ( { p } X. K ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> a = b ) ) )
92	57, 91	dom2lem 7593	. . . . 5 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-> _V )
93		f1f1orn 5810	. . . . 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 ) ) )
94	92, 93	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 ) ) )
95		fveq2 5849	. . . . . 6 |- ( a = z -> ( ^ ` a ) = ( ^ ` z ) )
96		eqid 2402	. . . . . 6 |- ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) = ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) )
97	95, 96, 1	fvmpt 5932	. . . . 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 464	. . . 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 5849	. . . . . . . . . . . . . . . 16 |- ( a = <. p , k >. -> ( ^ ` a ) = ( ^ ` <. p , k >. ) )
100	99, 4	syl6eqr 2461	. . . . . . . . . . . . . . 15 |- ( a = <. p , k >. -> ( ^ ` a ) = ( p ^ k ) )
101	100	eleq1d 2471	. . . . . . . . . . . . . 14 |- ( a = <. p , k >. -> ( ( ^ ` a ) e. A <-> ( p ^ k ) e. A ) )
102	37, 101	syl5ibrcom 222	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( a = <. p , k >. -> ( ^ ` a ) e. A ) )
103	102	impancom 438	. . . . . . . . . . . 12 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> ( ^ ` a ) e. A ) )
104	103	expimpd 601	. . . . . . . . . . 11 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
105	104	exlimdvv 1746	. . . . . . . . . 10 |- ( ph -> ( E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
106	58, 105	syl5bi 217	. . . . . . . . 9 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. A ) )
107	106	imp 427	. . . . . . . 8 |- ( ( ph /\ a e. U_ p e. P ( { p } X. K ) ) -> ( ^ ` a ) e. A )
108	107, 96	fmptd 6033	. . . . . . 7 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) --> A )
109		frn 5720	. . . . . . 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 3442	. . . . 5 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> y e. A )
112	46	nfel1 2580	. . . . . . 7 |- F/ x [_ y / x ]_ B e. CC
113	47	eleq1d 2471	. . . . . . 7 |- ( x = y -> ( B e. CC <-> [_ y / x ]_ B e. CC ) )
114	112, 113	rspc 3154	. . . . . 6 |- ( y e. A -> ( A. x e. A B e. CC -> [_ y / x ]_ B e. CC ) )
115	39, 114	mpan9 467	. . . . 5 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. CC )
116	111, 115	syldan 468	. . . 4 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> [_ y / x ]_ B e. CC )
117	49, 55, 94, 98, 116	fsumf1o 13694	. . 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	48, 117	syl5eq 2455	. 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 3442	. . . 4 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> x e. A )
120	119, 38	syldan 468	. . 3 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B e. CC )
121		eldif 3424	. . . . 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	96, 56	elrnmpti 5074	. . . . . . . . . 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 2416	. . . . . . . . . . 11 |- ( a = <. p , k >. -> ( x = ( ^ ` a ) <-> x = ( p ^ k ) ) )
124	123	rexiunxp 4964	. . . . . . . . . 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 249	. . . . . . . . 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 459	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> x = ( p ^ k ) )
127		simplr 754	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> x e. A )
128	126, 127	eqeltrrd 2491	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( p ^ k ) e. A )
129	14	rbaibd 911	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( p ^ k ) e. A ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
130	129	adantlr 713	. . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
132	131	pm5.32da 639	. . . . . . . . . . . . 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 448	. . . . . . . . . . . . 13 |- ( ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. P /\ k e. K ) ) )
134		ancom 448	. . . . . . . . . . . . 13 |- ( ( ( p e. Prime /\ k e. NN ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. Prime /\ k e. NN ) ) )
135	132, 133, 134	3bitr4g 288	. . . . . . . . . . . 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 1737	. . . . . . . . . . 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 2930	. . . . . . . . . . 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 2930	. . . . . . . . . . 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 288	. . . . . . . . . 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 3442	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. NN )
142		isppw2 23770	. . . . . . . . . . 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 256	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( E. p e. P E. k e. K x = ( p ^ k ) <-> (Λ ` x ) =/= 0 ) )
145	125, 144	syl5bb 257	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) <-> (Λ ` x ) =/= 0 ) )
146	145	necon2bbid 2659	. . . . . . 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 639	. . . . . 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 432	. . . . . 6 |- ( ph -> ( ( x e. A /\ (Λ ` x ) = 0 ) -> B = 0 ) )
150	147, 149	sylbird 235	. . . . 5 |- ( ph -> ( ( x e. A /\ -. x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B = 0 ) )
151	121, 150	syl5bi 217	. . . 4 |- ( ph -> ( x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B = 0 ) )
152	151	imp 427	. . 3 |- ( ( ph /\ x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) ) -> B = 0 )
153	110, 120, 152, 12	fsumss 13696	. 2 |- ( ph -> sum_ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) B = sum_ x e. A B )
154	44, 118, 153	3eqtr2rd 2450	1 |- ( ph -> sum_ x e. A B = sum_ p e. P sum_ k e. K C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   E.wex 1633   e. wcel 1842   =/= wne 2598   A.wral 2754   E.wrex 2755   _Vcvv 3059   [_csb 3373   \ cdif 3411   C_ wss 3414   {csn 3972   <.cop 3978   U_ciun 4271   |-> cmpt 4453   X. cxp 4821   ran crn 4824   -->wf 5565   -1-1->wf1 5566   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   Fincfn 7554   CCcc 9520   0cc0 9522   NNcn 10576   ^cexp 12210   sum_csu 13657   Primecprime 14426  Λcvma 23746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-dvds 14196  df-gcd 14354  df-prm 14427  df-pc 14570  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-vma 23752

This theorem is referenced by:  fsumvma2  23870  vmasum  23872