Theorem fsumvma 23211

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 5869	. . . . 5 |- ( ^ ` z ) e. _V
2	1	a1i 11	. . . 4 |- ( z = <. p , k >. -> ( ^ ` z ) e. _V )
3		fveq2 5859	. . . . . . . 8 |- ( z = <. p , k >. -> ( ^ ` z ) = ( ^ ` <. p , k >. ) )
4		df-ov 6280	. . . . . . . 8 |- ( p ^ k ) = ( ^ ` <. p , k >. )
5	3, 4	syl6eqr 2521	. . . . . . 7 |- ( z = <. p , k >. -> ( ^ ` z ) = ( p ^ k ) )
6	5	eqeq2d 2476	. . . . . 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 3457	. . 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 3505	. . . . . . . . 9 |- ( ( ph /\ p e. P ) -> K C_ NN )
26	25	sselda 3499	. . . . . . . 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 2462	. . . . . . . 8 |- p = p
29		prmexpb 14108	. . . . . . . . 9 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> ( p = p /\ k = z ) ) )
30	29	baibd 902	. . . . . . . 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 1224	. . . . . 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 7547	. . . 4 |- ( ( ph /\ p e. P ) -> ( k e. K |-> ( p ^ k ) ) : K -1-1-> A )
35		f1fi 7798	. . . 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 2873	. . . . 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 2531	. . . . 5 |- ( x = ( p ^ k ) -> ( B e. CC <-> C e. CC ) )
42	41	rspcv 3205	. . . 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 13537	. 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 2624	. . . 4 |- F/_ y B
46		nfcsb1v 3446	. . . 4 |- F/_ x [_ y / x ]_ B
47		csbeq1a 3439	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
48	45, 46, 47	cbvsumi 13470	. . 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 3433	. . . 4 |- ( y = ( ^ ` z ) -> [_ y / x ]_ B = [_ ( ^ ` z ) / x ]_ B )
50		snfi 7588	. . . . . . 7 |- { p } e. Fin
51		xpfi 7782	. . . . . . 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 2873	. . . . 5 |- ( ph -> A. p e. P ( { p } X. K ) e. Fin )
54		iunfi 7799	. . . . 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 5869	. . . . . . 7 |- ( ^ ` a ) e. _V
57	56	a1ii 27	. . . . . 6 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. _V ) )
58		eliunxp 5133	. . . . . . . . 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 5023	. . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . 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 1696	. . . . . . . . 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 3505	. . . . . . . . 9 |- ( ph -> U_ p e. P ( { p } X. K ) C_ ( Prime X. NN ) )
69	68	sseld 3498	. . . . . . . 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 6813	. . . . . . . . . . 11 |- ( a e. ( Prime X. NN ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
72	71	fveq2d 5863	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
73		df-ov 6280	. . . . . . . . . 10 |- ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
74	72, 73	syl6eqr 2521	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ( 1st ` a ) ^ ( 2nd ` a ) ) )
75		1st2nd2 6813	. . . . . . . . . . 11 |- ( b e. ( Prime X. NN ) -> b = <. ( 1st ` b ) , ( 2nd ` b ) >. )
76	75	fveq2d 5863	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. ) )
77		df-ov 6280	. . . . . . . . . 10 |- ( ( 1st ` b ) ^ ( 2nd ` b ) ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. )
78	76, 77	syl6eqr 2521	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) )
79	74, 78	eqeqan12d 2485	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) ) )
80		xp1st 6806	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 1st ` a ) e. Prime )
81		xp2nd 6807	. . . . . . . . . 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 6806	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 1st ` b ) e. Prime )
84		xp2nd 6807	. . . . . . . . . 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 14108	. . . . . . . . . 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 823	. . . . . . . . 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 6815	. . . . . . . 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 7547	. . . . 5 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-> _V )
93		f1f1orn 5820	. . . . 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 5859	. . . . . 6 |- ( a = z -> ( ^ ` a ) = ( ^ ` z ) )
96		eqid 2462	. . . . . 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 5943	. . . . 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 5859	. . . . . . . . . . . . . . . 16 |- ( a = <. p , k >. -> ( ^ ` a ) = ( ^ ` <. p , k >. ) )
100	99, 4	syl6eqr 2521	. . . . . . . . . . . . . . 15 |- ( a = <. p , k >. -> ( ^ ` a ) = ( p ^ k ) )
101	100	eleq1d 2531	. . . . . . . . . . . . . 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 1696	. . . . . . . . . 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 6038	. . . . . . 7 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) --> A )
109		frn 5730	. . . . . . 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 3499	. . . . 5 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> y e. A )
112	46	nfel1 2640	. . . . . . 7 |- F/ x [_ y / x ]_ B e. CC
113	47	eleq1d 2531	. . . . . . 7 |- ( x = y -> ( B e. CC <-> [_ y / x ]_ B e. CC ) )
114	112, 113	rspc 3203	. . . . . 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 13496	. . 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 2515	. 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 3499	. . . 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 3481	. . . . 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 5246	. . . . . . . . . 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 2476	. . . . . . . . . . 11 |- ( a = <. p , k >. -> ( x = ( ^ ` a ) <-> x = ( p ^ k ) ) )
124	123	rexiunxp 5136	. . . . . . . . . 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 2551	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( p ^ k ) e. A )
129	14	rbaibd 903	. . . . . . . . . . . . . . . 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 1687	. . . . . . . . . . 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 2980	. . . . . . . . . . 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 2980	. . . . . . . . . . 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 3499	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. NN )
142		isppw2 23112	. . . . . . . . . . 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 2718	. . . . . . 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 13498	. 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 2510	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 1374   E.wex 1591   e. wcel 1762   =/= wne 2657   A.wral 2809   E.wrex 2810   _Vcvv 3108   [_csb 3430   \ cdif 3468   C_ wss 3471   {csn 4022   <.cop 4028   U_ciun 4320   |-> cmpt 4500   X. cxp 4992   ran crn 4995   -->wf 5577   -1-1->wf1 5578   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775   Fincfn 7508   CCcc 9481   0cc0 9483   NNcn 10527   ^cexp 12124   sum_csu 13459   Primecprime 14067  Λcvma 23088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-vma 23094

This theorem is referenced by:  fsumvma2  23212  vmasum  23214