Theorem fsumvma 24078

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 5830	. . . . 5 |- ( ^ ` z ) e. _V
2	1	a1i 11	. . . 4 |- ( z = <. p , k >. -> ( ^ ` z ) e. _V )
3		fveq2 5820	. . . . . . . 8 |- ( z = <. p , k >. -> ( ^ ` z ) = ( ^ ` <. p , k >. ) )
4		df-ov 6247	. . . . . . . 8 |- ( p ^ k ) = ( ^ ` <. p , k >. )
5	3, 4	syl6eqr 2475	. . . . . . 7 |- ( z = <. p , k >. -> ( ^ ` z ) = ( p ^ k ) )
6	5	eqeq2d 2433	. . . . . 6 |- ( z = <. p , k >. -> ( x = ( ^ ` z ) <-> x = ( p ^ k ) ) )
7	6	biimpa 486	. . . . 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 3360	. . 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 466	. . . 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 210	. . . . . . . 8 |- ( ph -> ( ( p e. P /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) ) )
16	15	impl 624	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. A ) )
17	16	simprd 464	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p ^ k ) e. A )
18	17	ex 435	. . . . 5 |- ( ( ph /\ p e. P ) -> ( k e. K -> ( p ^ k ) e. A ) )
19	16	simpld 460	. . . . . . . . 9 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> ( p e. Prime /\ k e. NN ) )
20	19	simpld 460	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> p e. Prime )
21	20	adantrr 721	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> p e. Prime )
22	19	simprd 464	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ k e. K ) -> k e. NN )
23	22	adantrr 721	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> k e. NN )
24	22	ex 435	. . . . . . . . . 10 |- ( ( ph /\ p e. P ) -> ( k e. K -> k e. NN ) )
25	24	ssrdv 3408	. . . . . . . . 9 |- ( ( ph /\ p e. P ) -> K C_ NN )
26	25	sselda 3402	. . . . . . . 8 |- ( ( ( ph /\ p e. P ) /\ z e. K ) -> z e. NN )
27	26	adantrl 720	. . . . . . 7 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> z e. NN )
28		eqid 2423	. . . . . . . 8 |- p = p
29		prmexpb 14608	. . . . . . . . 9 |- ( ( ( p e. Prime /\ p e. Prime ) /\ ( k e. NN /\ z e. NN ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> ( p = p /\ k = z ) ) )
30	29	baibd 917	. . . . . . . 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 675	. . . . . . 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 1265	. . . . . 6 |- ( ( ( ph /\ p e. P ) /\ ( k e. K /\ z e. K ) ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) )
33	32	ex 435	. . . . 5 |- ( ( ph /\ p e. P ) -> ( ( k e. K /\ z e. K ) -> ( ( p ^ k ) = ( p ^ z ) <-> k = z ) ) )
34	18, 33	dom2lem 7558	. . . 4 |- ( ( ph /\ p e. P ) -> ( k e. K |-> ( p ^ k ) ) : K -1-1-> A )
35		f1fi 7809	. . . 4 |- ( ( A e. Fin /\ ( k e. K |-> ( p ^ k ) ) : K -1-1-> A ) -> K e. Fin )
36	13, 34, 35	syl2anc 665	. . 3 |- ( ( ph /\ p e. P ) -> K e. Fin )
37	14	simplbda 628	. . . 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 2774	. . . . 5 |- ( ph -> A. x e. A B e. CC )
40	39	adantr 466	. . . 4 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> A. x e. A B e. CC )
41	8	eleq1d 2485	. . . . 5 |- ( x = ( p ^ k ) -> ( B e. CC <-> C e. CC ) )
42	41	rspcv 3116	. . . 4 |- ( ( p ^ k ) e. A -> ( A. x e. A B e. CC -> C e. CC ) )
43	37, 40, 42	sylc 62	. . 3 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> C e. CC )
44	10, 11, 36, 43	fsum2d 13770	. 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 3349	. . . 4 |- F/_ x [_ y / x ]_ B
47		csbeq1a 3342	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
48	45, 46, 47	cbvsumi 13701	. . 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 3336	. . . 4 |- ( y = ( ^ ` z ) -> [_ y / x ]_ B = [_ ( ^ ` z ) / x ]_ B )
50		snfi 7599	. . . . . . 7 |- { p } e. Fin
51		xpfi 7790	. . . . . . 7 |- ( ( { p } e. Fin /\ K e. Fin ) -> ( { p } X. K ) e. Fin )
52	50, 36, 51	sylancr 667	. . . . . 6 |- ( ( ph /\ p e. P ) -> ( { p } X. K ) e. Fin )
53	52	ralrimiva 2774	. . . . 5 |- ( ph -> A. p e. P ( { p } X. K ) e. Fin )
54		iunfi 7810	. . . . 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 665	. . . 4 |- ( ph -> U_ p e. P ( { p } X. K ) e. Fin )
56		fvex 5830	. . . . . . 7 |- ( ^ ` a ) e. _V
57	56	2a1i 12	. . . . . 6 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. _V ) )
58		eliunxp 4929	. . . . . . . . 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 627	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( p e. Prime /\ k e. NN ) )
60		opelxp 4821	. . . . . . . . . . . . . 14 |- ( <. p , k >. e. ( Prime X. NN ) <-> ( p e. Prime /\ k e. NN ) )
61	59, 60	sylibr 215	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> <. p , k >. e. ( Prime X. NN ) )
62		eleq1 2489	. . . . . . . . . . . . 13 |- ( a = <. p , k >. -> ( a e. ( Prime X. NN ) <-> <. p , k >. e. ( Prime X. NN ) ) )
63	61, 62	syl5ibrcom 225	. . . . . . . . . . . 12 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( a = <. p , k >. -> a e. ( Prime X. NN ) ) )
64	63	impancom 441	. . . . . . . . . . 11 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> a e. ( Prime X. NN ) ) )
65	64	expimpd 606	. . . . . . . . . 10 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
66	65	exlimdvv 1773	. . . . . . . . 9 |- ( ph -> ( E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> a e. ( Prime X. NN ) ) )
67	58, 66	syl5bi 220	. . . . . . . 8 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> a e. ( Prime X. NN ) ) )
68	67	ssrdv 3408	. . . . . . . . 9 |- ( ph -> U_ p e. P ( { p } X. K ) C_ ( Prime X. NN ) )
69	68	sseld 3401	. . . . . . . 8 |- ( ph -> ( b e. U_ p e. P ( { p } X. K ) -> b e. ( Prime X. NN ) ) )
70	67, 69	anim12d 565	. . . . . . 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 6783	. . . . . . . . . . 11 |- ( a e. ( Prime X. NN ) -> a = <. ( 1st ` a ) , ( 2nd ` a ) >. )
72	71	fveq2d 5824	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. ) )
73		df-ov 6247	. . . . . . . . . 10 |- ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ^ ` <. ( 1st ` a ) , ( 2nd ` a ) >. )
74	72, 73	syl6eqr 2475	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ^ ` a ) = ( ( 1st ` a ) ^ ( 2nd ` a ) ) )
75		1st2nd2 6783	. . . . . . . . . . 11 |- ( b e. ( Prime X. NN ) -> b = <. ( 1st ` b ) , ( 2nd ` b ) >. )
76	75	fveq2d 5824	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. ) )
77		df-ov 6247	. . . . . . . . . 10 |- ( ( 1st ` b ) ^ ( 2nd ` b ) ) = ( ^ ` <. ( 1st ` b ) , ( 2nd ` b ) >. )
78	76, 77	syl6eqr 2475	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ^ ` b ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) )
79	74, 78	eqeqan12d 2439	. . . . . . . 8 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> ( ( 1st ` a ) ^ ( 2nd ` a ) ) = ( ( 1st ` b ) ^ ( 2nd ` b ) ) ) )
80		xp1st 6776	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 1st ` a ) e. Prime )
81		xp2nd 6777	. . . . . . . . . 10 |- ( a e. ( Prime X. NN ) -> ( 2nd ` a ) e. NN )
82	80, 81	jca 534	. . . . . . . . 9 |- ( a e. ( Prime X. NN ) -> ( ( 1st ` a ) e. Prime /\ ( 2nd ` a ) e. NN ) )
83		xp1st 6776	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 1st ` b ) e. Prime )
84		xp2nd 6777	. . . . . . . . . 10 |- ( b e. ( Prime X. NN ) -> ( 2nd ` b ) e. NN )
85	83, 84	jca 534	. . . . . . . . 9 |- ( b e. ( Prime X. NN ) -> ( ( 1st ` b ) e. Prime /\ ( 2nd ` b ) e. NN ) )
86		prmexpb 14608	. . . . . . . . . 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 833	. . . . . . . . 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 479	. . . . . . . 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 6785	. . . . . . . 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 282	. . . . . . 7 |- ( ( a e. ( Prime X. NN ) /\ b e. ( Prime X. NN ) ) -> ( ( ^ ` a ) = ( ^ ` b ) <-> a = b ) )
91	70, 90	syl6 34	. . . . . 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 7558	. . . . 5 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) -1-1-> _V )
93		f1f1orn 5780	. . . . 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 5820	. . . . . 6 |- ( a = z -> ( ^ ` a ) = ( ^ ` z ) )
96		eqid 2423	. . . . . 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 5903	. . . . 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 467	. . . 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 5820	. . . . . . . . . . . . . . . 16 |- ( a = <. p , k >. -> ( ^ ` a ) = ( ^ ` <. p , k >. ) )
100	99, 4	syl6eqr 2475	. . . . . . . . . . . . . . 15 |- ( a = <. p , k >. -> ( ^ ` a ) = ( p ^ k ) )
101	100	eleq1d 2485	. . . . . . . . . . . . . 14 |- ( a = <. p , k >. -> ( ( ^ ` a ) e. A <-> ( p ^ k ) e. A ) )
102	37, 101	syl5ibrcom 225	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p e. P /\ k e. K ) ) -> ( a = <. p , k >. -> ( ^ ` a ) e. A ) )
103	102	impancom 441	. . . . . . . . . . . 12 |- ( ( ph /\ a = <. p , k >. ) -> ( ( p e. P /\ k e. K ) -> ( ^ ` a ) e. A ) )
104	103	expimpd 606	. . . . . . . . . . 11 |- ( ph -> ( ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
105	104	exlimdvv 1773	. . . . . . . . . 10 |- ( ph -> ( E. p E. k ( a = <. p , k >. /\ ( p e. P /\ k e. K ) ) -> ( ^ ` a ) e. A ) )
106	58, 105	syl5bi 220	. . . . . . . . 9 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) -> ( ^ ` a ) e. A ) )
107	106	imp 430	. . . . . . . 8 |- ( ( ph /\ a e. U_ p e. P ( { p } X. K ) ) -> ( ^ ` a ) e. A )
108	107, 96	fmptd 6000	. . . . . . 7 |- ( ph -> ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) : U_ p e. P ( { p } X. K ) --> A )
109		frn 5690	. . . . . . 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 3402	. . . . 5 |- ( ( ph /\ y e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> y e. A )
112	46	nfel1 2578	. . . . . . 7 |- F/ x [_ y / x ]_ B e. CC
113	47	eleq1d 2485	. . . . . . 7 |- ( x = y -> ( B e. CC <-> [_ y / x ]_ B e. CC ) )
114	112, 113	rspc 3114	. . . . . 6 |- ( y e. A -> ( A. x e. A B e. CC -> [_ y / x ]_ B e. CC ) )
115	39, 114	mpan9 471	. . . . 5 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. CC )
116	111, 115	syldan 472	. . . 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 13727	. . 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 2469	. 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 3402	. . . 4 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> x e. A )
120	119, 38	syldan 472	. . 3 |- ( ( ph /\ x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B e. CC )
121		eldif 3384	. . . . 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 5042	. . . . . . . . . 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 2433	. . . . . . . . . . 11 |- ( a = <. p , k >. -> ( x = ( ^ ` a ) <-> x = ( p ^ k ) ) )
124	123	rexiunxp 4932	. . . . . . . . . 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 252	. . . . . . . . 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 462	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> x = ( p ^ k ) )
127		simplr 760	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> x e. A )
128	126, 127	eqeltrrd 2502	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( p ^ k ) e. A )
129	14	rbaibd 918	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( p ^ k ) e. A ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
130	129	adantlr 719	. . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. A ) /\ x = ( p ^ k ) ) -> ( ( p e. P /\ k e. K ) <-> ( p e. Prime /\ k e. NN ) ) )
132	131	pm5.32da 645	. . . . . . . . . . . . 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 451	. . . . . . . . . . . . 13 |- ( ( ( p e. P /\ k e. K ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. P /\ k e. K ) ) )
134		ancom 451	. . . . . . . . . . . . 13 |- ( ( ( p e. Prime /\ k e. NN ) /\ x = ( p ^ k ) ) <-> ( x = ( p ^ k ) /\ ( p e. Prime /\ k e. NN ) ) )
135	132, 133, 134	3bitr4g 291	. . . . . . . . . . . 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 1764	. . . . . . . . . . 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 2885	. . . . . . . . . . 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 2885	. . . . . . . . . . 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 291	. . . . . . . . . 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 3402	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. NN )
142		isppw2 23979	. . . . . . . . . . 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 259	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( E. p e. P E. k e. K x = ( p ^ k ) <-> (Λ ` x ) =/= 0 ) )
145	125, 144	syl5bb 260	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) <-> (Λ ` x ) =/= 0 ) )
146	145	necon2bbid 2639	. . . . . . 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 645	. . . . . 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 435	. . . . . 6 |- ( ph -> ( ( x e. A /\ (Λ ` x ) = 0 ) -> B = 0 ) )
150	147, 149	sylbird 238	. . . . 5 |- ( ph -> ( ( x e. A /\ -. x e. ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B = 0 ) )
151	121, 150	syl5bi 220	. . . 4 |- ( ph -> ( x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) -> B = 0 ) )
152	151	imp 430	. . 3 |- ( ( ph /\ x e. ( A \ ran ( a e. U_ p e. P ( { p } X. K ) |-> ( ^ ` a ) ) ) ) -> B = 0 )
153	110, 120, 152, 12	fsumss 13729	. 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 2464	1 |- ( ph -> sum_ x e. A B = sum_ p e. P sum_ k e. K C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   =/= wne 2594   A.wral 2709   E.wrex 2710   _Vcvv 3017   [_csb 3333   \ cdif 3371   C_ wss 3374   {csn 3936   <.cop 3942   U_ciun 4237   |-> cmpt 4420   X. cxp 4789   ran crn 4792   -->wf 5535   -1-1->wf1 5536   -1-1-onto->wf1o 5538   ` cfv 5539  (class class class)co 6244   1stc1st 6744   2ndc2nd 6745   Fincfn 7519   CCcc 9483   0cc0 9485   NNcn 10555   ^cexp 12217   sum_csu 13690   Primecprime 14560  Λcvma 23955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563  ax-addf 9564  ax-mulf 9565

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-supp 6865  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7473  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-fsupp 7832  df-fi 7873  df-sup 7904  df-inf 7905  df-oi 7973  df-card 8320  df-cda 8544  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-9 10621  df-10 10622  df-n0 10816  df-z 10884  df-dec 10998  df-uz 11106  df-q 11211  df-rp 11249  df-xneg 11355  df-xadd 11356  df-xmul 11357  df-ioo 11585  df-ioc 11586  df-ico 11587  df-icc 11588  df-fz 11731  df-fzo 11862  df-fl 11973  df-mod 12042  df-seq 12159  df-exp 12218  df-fac 12405  df-bc 12433  df-hash 12461  df-shft 13069  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-limsup 13464  df-clim 13490  df-rlim 13491  df-sum 13691  df-ef 14059  df-sin 14061  df-cos 14062  df-pi 14064  df-dvds 14244  df-gcd 14407  df-prm 14561  df-pc 14725  df-struct 15061  df-ndx 15062  df-slot 15063  df-base 15064  df-sets 15065  df-ress 15066  df-plusg 15141  df-mulr 15142  df-starv 15143  df-sca 15144  df-vsca 15145  df-ip 15146  df-tset 15147  df-ple 15148  df-ds 15150  df-unif 15151  df-hom 15152  df-cco 15153  df-rest 15259  df-topn 15260  df-0g 15278  df-gsum 15279  df-topgen 15280  df-pt 15281  df-prds 15284  df-xrs 15338  df-qtop 15344  df-imas 15345  df-xps 15348  df-mre 15430  df-mrc 15431  df-acs 15433  df-mgm 16426  df-sgrp 16465  df-mnd 16475  df-submnd 16521  df-mulg 16614  df-cntz 16909  df-cmn 17370  df-psmet 18900  df-xmet 18901  df-met 18902  df-bl 18903  df-mopn 18904  df-fbas 18905  df-fg 18906  df-cnfld 18909  df-top 19858  df-bases 19859  df-topon 19860  df-topsp 19861  df-cld 19971  df-ntr 19972  df-cls 19973  df-nei 20051  df-lp 20089  df-perf 20090  df-cn 20180  df-cnp 20181  df-haus 20268  df-tx 20514  df-hmeo 20707  df-fil 20798  df-fm 20890  df-flim 20891  df-flf 20892  df-xms 21272  df-ms 21273  df-tms 21274  df-cncf 21847  df-limc 22758  df-dv 22759  df-log 23443  df-vma 23961

This theorem is referenced by:  fsumvma2  24079  vmasum  24081