Theorem dvdsflsumcom 24914

Description: A sum commutation from sum_ n <_ A , sum_ d || n , B ( n , d ) to sum_ d <_ A , sum_ m <_ A / d , B ( n , d m ). (Contributed by Mario Carneiro, 4-May-2016.)

Hypotheses

Ref	Expression
dvdsflsumcom.1	|- ( n = ( d x. m ) -> B = C )
dvdsflsumcom.2	|- ( ph -> A e. RR )
dvdsflsumcom.3	|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ d e. { x e. NN | x || n } ) ) -> B e. CC )

Assertion

Ref	Expression
dvdsflsumcom	|- ( ph -> sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ d e. { x e. NN | x || n } B = sum_ d e. ( 1 ... ( |_ ` A ) ) sum_ m e. ( 1 ... ( |_ ` ( A / d ) ) ) C )

Distinct variable groups:   m, d, n, x, A   B, m   C, n   ph, d, m, n

Allowed substitution hints:   ph( x)   B( x, n, d)   C( x, m, d)

Proof of Theorem dvdsflsumcom

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12772	. . 3 |- ( ph -> ( 1 ... ( |_ ` A ) ) e. Fin )
2		fzfid 12772	. . . 4 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 ... n ) e. Fin )
3		elfznn 12370	. . . . . 6 |- ( n e. ( 1 ... ( |_ ` A ) ) -> n e. NN )
4	3	adantl 482	. . . . 5 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> n e. NN )
5		dvdsssfz1 15040	. . . . 5 |- ( n e. NN -> { x e. NN | x || n } C_ ( 1 ... n ) )
6	4, 5	syl 17	. . . 4 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> { x e. NN | x || n } C_ ( 1 ... n ) )
7		ssfi 8180	. . . 4 |- ( ( ( 1 ... n ) e. Fin /\ { x e. NN | x || n } C_ ( 1 ... n ) ) -> { x e. NN | x || n } e. Fin )
8	2, 6, 7	syl2anc 693	. . 3 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> { x e. NN | x || n } e. Fin )
9		nnre 11027	. . . . . . . . . . . 12 |- ( d e. NN -> d e. RR )
10	9	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> d e. RR )
11	4	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> n e. NN )
12	11	nnred 11035	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> n e. RR )
13		dvdsflsumcom.2	. . . . . . . . . . . 12 |- ( ph -> A e. RR )
14	13	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> A e. RR )
15		nnz 11399	. . . . . . . . . . . . 13 |- ( d e. NN -> d e. ZZ )
16		dvdsle 15032	. . . . . . . . . . . . 13 |- ( ( d e. ZZ /\ n e. NN ) -> ( d || n -> d <_ n ) )
17	15, 4, 16	syl2anr 495	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ d e. NN ) -> ( d || n -> d <_ n ) )
18	17	impr 649	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> d <_ n )
19		fznnfl 12661	. . . . . . . . . . . . . 14 |- ( A e. RR -> ( n e. ( 1 ... ( |_ ` A ) ) <-> ( n e. NN /\ n <_ A ) ) )
20	13, 19	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( n e. ( 1 ... ( |_ ` A ) ) <-> ( n e. NN /\ n <_ A ) ) )
21	20	simplbda 654	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> n <_ A )
22	21	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> n <_ A )
23	10, 12, 14, 18, 22	letrd 10194	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ ( d e. NN /\ d || n ) ) -> d <_ A )
24	23	ex 450	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( d e. NN /\ d || n ) -> d <_ A ) )
25	24	pm4.71rd 667	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( d e. NN /\ d || n ) <-> ( d <_ A /\ ( d e. NN /\ d || n ) ) ) )
26		ancom 466	. . . . . . . . 9 |- ( ( d <_ A /\ ( d e. NN /\ d || n ) ) <-> ( ( d e. NN /\ d || n ) /\ d <_ A ) )
27		an32 839	. . . . . . . . 9 |- ( ( ( d e. NN /\ d || n ) /\ d <_ A ) <-> ( ( d e. NN /\ d <_ A ) /\ d || n ) )
28	26, 27	bitri 264	. . . . . . . 8 |- ( ( d <_ A /\ ( d e. NN /\ d || n ) ) <-> ( ( d e. NN /\ d <_ A ) /\ d || n ) )
29	25, 28	syl6bb 276	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( d e. NN /\ d || n ) <-> ( ( d e. NN /\ d <_ A ) /\ d || n ) ) )
30		fznnfl 12661	. . . . . . . . . 10 |- ( A e. RR -> ( d e. ( 1 ... ( |_ ` A ) ) <-> ( d e. NN /\ d <_ A ) ) )
31	13, 30	syl 17	. . . . . . . . 9 |- ( ph -> ( d e. ( 1 ... ( |_ ` A ) ) <-> ( d e. NN /\ d <_ A ) ) )
32	31	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( d e. ( 1 ... ( |_ ` A ) ) <-> ( d e. NN /\ d <_ A ) ) )
33	32	anbi1d 741	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( d e. ( 1 ... ( |_ ` A ) ) /\ d || n ) <-> ( ( d e. NN /\ d <_ A ) /\ d || n ) ) )
34	29, 33	bitr4d 271	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( d e. NN /\ d || n ) <-> ( d e. ( 1 ... ( |_ ` A ) ) /\ d || n ) ) )
35	34	pm5.32da 673	. . . . 5 |- ( ph -> ( ( n e. ( 1 ... ( |_ ` A ) ) /\ ( d e. NN /\ d || n ) ) <-> ( n e. ( 1 ... ( |_ ` A ) ) /\ ( d e. ( 1 ... ( |_ ` A ) ) /\ d || n ) ) ) )
36		an12 838	. . . . 5 |- ( ( n e. ( 1 ... ( |_ ` A ) ) /\ ( d e. ( 1 ... ( |_ ` A ) ) /\ d || n ) ) <-> ( d e. ( 1 ... ( |_ ` A ) ) /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ d || n ) ) )
37	35, 36	syl6bb 276	. . . 4 |- ( ph -> ( ( n e. ( 1 ... ( |_ ` A ) ) /\ ( d e. NN /\ d || n ) ) <-> ( d e. ( 1 ... ( |_ ` A ) ) /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ d || n ) ) ) )
38		breq1 4656	. . . . . 6 |- ( x = d -> ( x || n <-> d || n ) )
39	38	elrab 3363	. . . . 5 |- ( d e. { x e. NN | x || n } <-> ( d e. NN /\ d || n ) )
40	39	anbi2i 730	. . . 4 |- ( ( n e. ( 1 ... ( |_ ` A ) ) /\ d e. { x e. NN | x || n } ) <-> ( n e. ( 1 ... ( |_ ` A ) ) /\ ( d e. NN /\ d || n ) ) )
41		breq2 4657	. . . . . 6 |- ( x = n -> ( d || x <-> d || n ) )
42	41	elrab 3363	. . . . 5 |- ( n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } <-> ( n e. ( 1 ... ( |_ ` A ) ) /\ d || n ) )
43	42	anbi2i 730	. . . 4 |- ( ( d e. ( 1 ... ( |_ ` A ) ) /\ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } ) <-> ( d e. ( 1 ... ( |_ ` A ) ) /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ d || n ) ) )
44	37, 40, 43	3bitr4g 303	. . 3 |- ( ph -> ( ( n e. ( 1 ... ( |_ ` A ) ) /\ d e. { x e. NN | x || n } ) <-> ( d e. ( 1 ... ( |_ ` A ) ) /\ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } ) ) )
45		dvdsflsumcom.3	. . 3 |- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ d e. { x e. NN | x || n } ) ) -> B e. CC )
46	1, 1, 8, 44, 45	fsumcom2 14505	. 2 |- ( ph -> sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ d e. { x e. NN | x || n } B = sum_ d e. ( 1 ... ( |_ ` A ) ) sum_ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } B )
47		dvdsflsumcom.1	. . . 4 |- ( n = ( d x. m ) -> B = C )
48		fzfid 12772	. . . 4 |- ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 ... ( |_ ` ( A / d ) ) ) e. Fin )
49	13	adantr 481	. . . . 5 |- ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) -> A e. RR )
50	31	simprbda 653	. . . . 5 |- ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) -> d e. NN )
51		eqid 2622	. . . . 5 |- ( y e. ( 1 ... ( |_ ` ( A / d ) ) ) |-> ( d x. y ) ) = ( y e. ( 1 ... ( |_ ` ( A / d ) ) ) |-> ( d x. y ) )
52	49, 50, 51	dvdsflf1o 24913	. . . 4 |- ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) -> ( y e. ( 1 ... ( |_ ` ( A / d ) ) ) |-> ( d x. y ) ) : ( 1 ... ( |_ ` ( A / d ) ) ) -1-1-onto-> { x e. ( 1 ... ( |_ ` A ) ) | d || x } )
53		oveq2 6658	. . . . . 6 |- ( y = m -> ( d x. y ) = ( d x. m ) )
54		ovex 6678	. . . . . 6 |- ( d x. m ) e. _V
55	53, 51, 54	fvmpt 6282	. . . . 5 |- ( m e. ( 1 ... ( |_ ` ( A / d ) ) ) -> ( ( y e. ( 1 ... ( |_ ` ( A / d ) ) ) |-> ( d x. y ) ) ` m ) = ( d x. m ) )
56	55	adantl 482	. . . 4 |- ( ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) /\ m e. ( 1 ... ( |_ ` ( A / d ) ) ) ) -> ( ( y e. ( 1 ... ( |_ ` ( A / d ) ) ) |-> ( d x. y ) ) ` m ) = ( d x. m ) )
57	44	biimpar 502	. . . . . 6 |- ( ( ph /\ ( d e. ( 1 ... ( |_ ` A ) ) /\ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } ) ) -> ( n e. ( 1 ... ( |_ ` A ) ) /\ d e. { x e. NN | x || n } ) )
58	57, 45	syldan 487	. . . . 5 |- ( ( ph /\ ( d e. ( 1 ... ( |_ ` A ) ) /\ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } ) ) -> B e. CC )
59	58	anassrs 680	. . . 4 |- ( ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) /\ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } ) -> B e. CC )
60	47, 48, 52, 56, 59	fsumf1o 14454	. . 3 |- ( ( ph /\ d e. ( 1 ... ( |_ ` A ) ) ) -> sum_ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } B = sum_ m e. ( 1 ... ( |_ ` ( A / d ) ) ) C )
61	60	sumeq2dv 14433	. 2 |- ( ph -> sum_ d e. ( 1 ... ( |_ ` A ) ) sum_ n e. { x e. ( 1 ... ( |_ ` A ) ) | d || x } B = sum_ d e. ( 1 ... ( |_ ` A ) ) sum_ m e. ( 1 ... ( |_ ` ( A / d ) ) ) C )
62	46, 61	eqtrd 2656	1 |- ( ph -> sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ d e. { x e. NN | x || n } B = sum_ d e. ( 1 ... ( |_ ` A ) ) sum_ m e. ( 1 ... ( |_ ` ( A / d ) ) ) C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   1c1 9937   x. cmul 9941   <_ cle 10075   / cdiv 10684   NNcn 11020   ZZcz 11377   ...cfz 12326   |_cfl 12591   sum_csu 14416   || cdvds 14983

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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984

This theorem is referenced by:  dchrmusum2  25183  dchrvmasumlem1  25184  dchrvmasum2lem  25185  dchrisum0  25209  mudivsum  25219  mulogsum  25221  mulog2sumlem2  25224  vmalogdivsum2  25227  selberglem3  25236  selberg  25237  selberg34r  25260  pntsval2  25265