Theorem fsumvma 22574

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 5722	. . . . 5 |- ( ^ ` z ) e. _V
2	1	a1i 11	. . . 4 |- ( z = <. p , k >. -> ( ^ ` z ) e. _V )
3		fveq2 5712	. . . . . . . 8 |- ( z = <. p , k >. -> ( ^ ` z ) = ( ^ ` <. p , k >. ) )
4		df-ov 6115	. . . . . . . 8 |- ( p ^ k ) = ( ^ ` <. p , k >. )
5	3, 4	syl6eqr 2493	. . . . . . 7 |- ( z = <. p , k >. -> ( ^ ` z ) = ( p ^ k ) )
6	5	eqeq2d 2454	. . . . . 6 |- ( z = <. p , k >. -> ( x = ( ^ ` z ) <-> x = ( p ^ k ) ) )
7	6	biimpa 484	. . . . 5 |- ( ( z = <. p , k >. /\ x = ( ^ ` z ) ) -> x = ( p ^ k ) )
8		fsumvma.1	. . . . 5 |- ( x = ( p ^ k ) -> B = C )
9	7, 8	syl 16	. . . 4 |- ( ( z = <. p , k >. /\ x = ( ^ ` z ) ) -> B = C )
10	2, 9	csbied 3335	. . 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 465	. . . 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 620	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) )
17	16	simprd 463	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p ^ k ) e. A )
18	17	ex 434	. . . . 5 |- ( ( ph /\ p e. P ) -> ( k e. K -> ( p ^ k ) e. A ) )
19	16	simpld 459	. . . . . . . . 9 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p e. Prime /\ k e. NN ) )
20	19	simpld 459	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> p e. Prime )
21	20	adantrr 716	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> p e. Prime )
22	19	simprd 463	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> k e. NN )
23	22	adantrr 716	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> k e. NN )
24	22	ex 434	. . . . . . . . . 10 |- ( ( ph /\ p e. P ) -> ( k e. K -> k e. NN ) )
25	24	ssrdv 3383	. . . . . . . . 9 |- ( ( ph /\ p e. P ) -> K C_ NN )
26	25	sselda 3377	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ z e. K ) -> z e. NN )
27	26	adantrl 715	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> z e. NN )
28		eqid 2443	. . . . . . . 8 |- p = p
29		prmexpb 13824	. . . . . . . . 9 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> ( p = p /\ k = z ) ) )
30	29	baibd 900	. . . . . . . 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 671	. . . . . . 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 1219	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
33	32	ex 434	. . . . 5 |- ( ( ph /\ p e. P ) -> ( ( k e. K /\ z e. K ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) ) )
34	18, 33	dom2lem 7370	. . . 4 |- ( ( ph /\ p e. P ) -> ( k e. K |-> ( p ^ k ) ) : K -1-1-> A )
35		f1fi 7619	. . . 4 |- ( ( A e. Fin /\ ( k e. K |-> ( p ^ k ) ) : K -1-1-> A ) -> K e. Fin )
36	13, 34, 35	syl2anc 661	. . 3 |- ( ( ph /\ p e. P ) -> K e. Fin )
37	14	simplbda 624	. . . 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 2820	. . . . 5 |- ( ph -> A. x e. A B e. CC )
40	39	adantr 465	. . . 4 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> A. x e. A B e. CC )
41	8	eleq1d 2509	. . . . 5 |- ( x = ( p ^ k ) -> ( B e. CC <-> C e. CC ) )
42	41	rspcv 3090	. . . 4 |- ( ( p ^ k ) e. A -> ( A. x e. A B e. CC -> C e. CC ) )
43	37, 40, 42	sylc 60	. . 3 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> C e. CC )
44	10, 11, 36, 43	fsum2d 13259	. 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 2589	. . . 4 |- F/_ y B
46		nfcsb1v 3325	. . . 4 |- F/_ x [_ y / x ]_ B
47		csbeq1a 3318	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
48	45, 46, 47	cbvsumi 13195	. . 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 3312	. . . 4 |- ( y = ( ^ ` z ) -> [_ y / x ]_ B = [_ ( ^ ` z ) / x ]_ B )
50		snfi 7411	. . . . . . 7 |- { p } e. Fin
51		xpfi 7604	. . . . . . 7 |- ( ( { p } e. Fin /\ K e. Fin ) -> ( { p } X. K ) e. Fin )
52	50, 36, 51	sylancr 663	. . . . . 6 |- ( ( ph /\ p e. P ) -> ( { p } X. K ) e. Fin )
53	52	ralrimiva 2820	. . . . 5 |- ( ph -> A. p e. P ( { p } X. K ) e. Fin )
54		iunfi 7620	. . . . 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 661	. . . 4 |- ( ph -> U_ p e. P ( { p } X. K ) e. Fin )
56		fvex 5722	. . . . . . 7 |- ( ^ ` a ) e. _V
57	56	a1ii 27	. . . . . 6 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. _V ) )
58		eliunxp 4998	. . . . . . . . 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 623	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( p e. Prime /\ k e. NN ) )
60		opelxp 4890	. . . . . . . . . . . . . 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 2503	. . . . . . . . . . . . 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 440	. . . . . . . . . . 11 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> a e. ( Prime X. NN ) ) )
65	64	expimpd 603	. . . . . . . . . 10 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
66	65	exlimdvv 1691	. . . . . . . . 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 3383	. . . . . . . . 9 |- ( ph -> U_ p e. P ( { p } X. K ) C_ ( Prime X. NN ) )
69	68	sseld 3376	. . . . . . . 8 |- ( ph -> ( b e. U_ p e. P ( { p } X. K ) -> b e. ( Prime X. NN ) ) )
70	67, 69	anim12d 563	. . . . . . 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 6634	. . . . . . . . . . 11 |- ( a e. ( Prime X. NN ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
72	71	fveq2d 5716	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
73		df-ov 6115	. . . . . . . . . 10 |- ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
74	72, 73	syl6eqr 2493	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ( 1st ` a ) ^ ( 2nd ` a ) ) )
75		1st2nd2 6634	. . . . . . . . . . 11 |- ( b e. ( Prime X. NN ) -> b = <. ( 1st ` b ) , ( 2nd ` b ) >. )
76	75	fveq2d 5716	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. ) )
77		df-ov 6115	. . . . . . . . . 10 |- ( ( 1st ` b ) ^ ( 2nd ` b ) ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. )
78	76, 77	syl6eqr 2493	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) )
79	74, 78	eqeqan12d 2458	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) ) )
80		xp1st 6627	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 1st ` a ) e. Prime )
81		xp2nd 6628	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 2nd ` a ) e. NN )
82	80, 81	jca 532	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ( 1st ` a ) e. Prime /\ ( 2nd ` a ) e. NN ) )
83		xp1st 6627	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 1st ` b ) e. Prime )
84		xp2nd 6628	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 2nd ` b ) e. NN )
85	83, 84	jca 532	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ( 1st ` b ) e. Prime /\ ( 2nd ` b ) e. NN ) )
86		prmexpb 13824	. . . . . . . . . 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 822	. . . . . . . . 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 477	. . . . . . . 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 6636	. . . . . . . 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 33	. . . . . 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 7370	. . . . 5 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-> _V )
93		f1f1orn 5673	. . . . 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 16	. . . 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 5712	. . . . . 6 |- ( a = z -> ( ^ ` a ) = ( ^ ` z ) )
96		eqid 2443	. . . . . 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 5795	. . . . 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 466	. . . 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 5712	. . . . . . . . . . . . . . . 16 |- ( a = <. p , k >. -> ( ^ ` a ) = ( ^ ` <. p , k >. ) )
100	99, 4	syl6eqr 2493	. . . . . . . . . . . . . . 15 |- ( a = <. p , k >. -> ( ^ ` a ) = ( p ^ k ) )
101	100	eleq1d 2509	. . . . . . . . . . . . . 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 440	. . . . . . . . . . . 12 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> ( ^ ` a ) e. A ) )
104	103	expimpd 603	. . . . . . . . . . 11 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
105	104	exlimdvv 1691	. . . . . . . . . 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 429	. . . . . . . 8 |- ( ( ph /\ a e. U_ p e. P ( { p } X. K ) ) -> ( ^ ` a ) e. A )
108	107, 96	fmptd 5888	. . . . . . 7 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) --> A )
109		frn 5586	. . . . . . 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 16	. . . . . 6 |- ( ph -> ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) C_ A )
111	110	sselda 3377	. . . . 5 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> y e. A )
112	46	nfel1 2604	. . . . . . 7 |- F/ x [_ y / x ]_ B e. CC
113	47	eleq1d 2509	. . . . . . 7 |- ( x = y -> ( B e. CC <-> [_ y / x ]_ B e. CC ) )
114	112, 113	rspc 3088	. . . . . 6 |- ( y e. A -> ( A. x e. A B e. CC -> [_ y / x ]_ B e. CC ) )
115	39, 114	mpan9 469	. . . . 5 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. CC )
116	111, 115	syldan 470	. . . 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 13221	. . 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 2487	. 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 3377	. . . 4 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> x e. A )
120	119, 38	syldan 470	. . 3 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B e. CC )
121		eldif 3359	. . . . 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 5111	. . . . . . . . . 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 2454	. . . . . . . . . . 11 |- ( a = <. p , k >. -> ( x = ( ^ ` a ) <-> x = ( p ^ k ) ) )
124	123	rexiunxp 5001	. . . . . . . . . 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 461	. . . . . . . . . . . . . . . 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 2518	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( p ^ k ) e. A )
129	14	rbaibd 901	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( p ^ k ) e. A ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
130	129	adantlr 714	. . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
132	131	pm5.32da 641	. . . . . . . . . . . . 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 450	. . . . . . . . . . . . 13 |- ( ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. P /\ k e. K ) ) )
134		ancom 450	. . . . . . . . . . . . 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 1682	. . . . . . . . . . 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 2774	. . . . . . . . . . 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 2774	. . . . . . . . . . 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 3377	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. NN )
142		isppw2 22475	. . . . . . . . . . 11 |- ( x e. NN -> ( (Λ ` x ) =/= 0 <-> E. p e. Prime E. k e. NN x = ( p ^ k ) ) )
143	141, 142	syl 16	. . . . . . . . . 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 2693	. . . . . . 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 641	. . . . . 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 434	. . . . . 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 429	. . 3 |- ( ( ph /\ x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) ) -> B = 0 )
153	110, 120, 152, 12	fsumss 13223	. 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 2482	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 369   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2620   A.wral 2736   E.wrex 2737   _Vcvv 2993   [_csb 3309   \ cdif 3346   C_ wss 3349   {csn 3898   <.cop 3904   U_ciun 4192   e. cmpt 4371   X. cxp 4859   ran crn 4862   -->wf 5435   -1-1->wf1 5436   -1-1-onto->wf1o 5438   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597   Fincfn 7331   CCcc 9301   0cc0 9303   NNcn 10343   ^cexp 11886   sum_csu 13184   Primecprime 13784  Λcvma 22451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-pi 13379  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-pt 14404  df-prds 14407  df-xrs 14461  df-qtop 14466  df-imas 14467  df-xps 14469  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cn 18853  df-cnp 18854  df-haus 18941  df-tx 19157  df-hmeo 19350  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-tms 19919  df-cncf 20476  df-limc 21363  df-dv 21364  df-log 22030  df-vma 22457

This theorem is referenced by:  fsumvma2  22575  vmasum  22577