Theorem ppidif 24889

Description: The difference of the prime-counting function π at two points counts the number of primes in an interval. (Contributed by Mario Carneiro, 21-Sep-2014.)

Assertion

Ref	Expression
ppidif	|- ( N e. ( ZZ>= ` M ) -> ( (π ` N ) - (π ` M ) ) = ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) )

Proof of Theorem ppidif

Step	Hyp	Ref	Expression
1		eluzelz 11697	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
2		eluzel2 11692	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3		2z 11409	. . . . . . 7 |- 2 e. ZZ
4		ifcl 4130	. . . . . . 7 |- ( ( M e. ZZ /\ 2 e. ZZ ) -> if ( M <_ 2 , M , 2 ) e. ZZ )
5	2, 3, 4	sylancl 694	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> if ( M <_ 2 , M , 2 ) e. ZZ )
6	3	a1i 11	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> 2 e. ZZ )
7	2	zred 11482	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. RR )
8		2re 11090	. . . . . . 7 |- 2 e. RR
9		min2 12021	. . . . . . 7 |- ( ( M e. RR /\ 2 e. RR ) -> if ( M <_ 2 , M , 2 ) <_ 2 )
10	7, 8, 9	sylancl 694	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> if ( M <_ 2 , M , 2 ) <_ 2 )
11		eluz2 11693	. . . . . 6 |- ( 2 e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) <-> ( if ( M <_ 2 , M , 2 ) e. ZZ /\ 2 e. ZZ /\ if ( M <_ 2 , M , 2 ) <_ 2 ) )
12	5, 6, 10, 11	syl3anbrc 1246	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> 2 e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) )
13		ppival2g 24855	. . . . 5 |- ( ( N e. ZZ /\ 2 e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) ) -> (π ` N ) = ( # ` ( ( if ( M <_ 2 , M , 2 ) ... N ) i^i Prime ) ) )
14	1, 12, 13	syl2anc 693	. . . 4 |- ( N e. ( ZZ>= ` M ) -> (π ` N ) = ( # ` ( ( if ( M <_ 2 , M , 2 ) ... N ) i^i Prime ) ) )
15		min1 12020	. . . . . . . . . . 11 |- ( ( M e. RR /\ 2 e. RR ) -> if ( M <_ 2 , M , 2 ) <_ M )
16	7, 8, 15	sylancl 694	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> if ( M <_ 2 , M , 2 ) <_ M )
17		eluz2 11693	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) <-> ( if ( M <_ 2 , M , 2 ) e. ZZ /\ M e. ZZ /\ if ( M <_ 2 , M , 2 ) <_ M ) )
18	5, 2, 16, 17	syl3anbrc 1246	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) )
19		id 22	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( ZZ>= ` M ) )
20		elfzuzb 12336	. . . . . . . . 9 |- ( M e. ( if ( M <_ 2 , M , 2 ) ... N ) <-> ( M e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) /\ N e. ( ZZ>= ` M ) ) )
21	18, 19, 20	sylanbrc 698	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( if ( M <_ 2 , M , 2 ) ... N ) )
22		fzsplit 12367	. . . . . . . 8 |- ( M e. ( if ( M <_ 2 , M , 2 ) ... N ) -> ( if ( M <_ 2 , M , 2 ) ... N ) = ( ( if ( M <_ 2 , M , 2 ) ... M ) u. ( ( M + 1 ) ... N ) ) )
23	21, 22	syl 17	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( if ( M <_ 2 , M , 2 ) ... N ) = ( ( if ( M <_ 2 , M , 2 ) ... M ) u. ( ( M + 1 ) ... N ) ) )
24	23	ineq1d 3813	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( if ( M <_ 2 , M , 2 ) ... N ) i^i Prime ) = ( ( ( if ( M <_ 2 , M , 2 ) ... M ) u. ( ( M + 1 ) ... N ) ) i^i Prime ) )
25		indir 3875	. . . . . 6 |- ( ( ( if ( M <_ 2 , M , 2 ) ... M ) u. ( ( M + 1 ) ... N ) ) i^i Prime ) = ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) )
26	24, 25	syl6eq 2672	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( if ( M <_ 2 , M , 2 ) ... N ) i^i Prime ) = ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) )
27	26	fveq2d 6195	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( # ` ( ( if ( M <_ 2 , M , 2 ) ... N ) i^i Prime ) ) = ( # ` ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
28	7	ltp1d 10954	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M < ( M + 1 ) )
29		fzdisj 12368	. . . . . . . 8 |- ( M < ( M + 1 ) -> ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) )
30	28, 29	syl 17	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) )
31	30	ineq1d 3813	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i ( ( M + 1 ) ... N ) ) i^i Prime ) = ( (/) i^i Prime ) )
32		inindir 3831	. . . . . 6 |- ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i ( ( M + 1 ) ... N ) ) i^i Prime ) = ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) )
33		0in 3969	. . . . . 6 |- ( (/) i^i Prime ) = (/)
34	31, 32, 33	3eqtr3g 2679	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) ) = (/) )
35		fzfi 12771	. . . . . . 7 |- ( if ( M <_ 2 , M , 2 ) ... M ) e. Fin
36		inss1 3833	. . . . . . 7 |- ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) C_ ( if ( M <_ 2 , M , 2 ) ... M )
37		ssfi 8180	. . . . . . 7 |- ( ( ( if ( M <_ 2 , M , 2 ) ... M ) e. Fin /\ ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) C_ ( if ( M <_ 2 , M , 2 ) ... M ) ) -> ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) e. Fin )
38	35, 36, 37	mp2an 708	. . . . . 6 |- ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) e. Fin
39		fzfi 12771	. . . . . . 7 |- ( ( M + 1 ) ... N ) e. Fin
40		inss1 3833	. . . . . . 7 |- ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N )
41		ssfi 8180	. . . . . . 7 |- ( ( ( ( M + 1 ) ... N ) e. Fin /\ ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N ) ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin )
42	39, 40, 41	mp2an 708	. . . . . 6 |- ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin
43		hashun 13171	. . . . . 6 |- ( ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) e. Fin /\ ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin /\ ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) ) = (/) ) -> ( # ` ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) = ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
44	38, 42, 43	mp3an12 1414	. . . . 5 |- ( ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) ) = (/) -> ( # ` ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) = ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
45	34, 44	syl 17	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( # ` ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) = ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
46	14, 27, 45	3eqtrd 2660	. . 3 |- ( N e. ( ZZ>= ` M ) -> (π ` N ) = ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
47		ppival2g 24855	. . . 4 |- ( ( M e. ZZ /\ 2 e. ( ZZ>= ` if ( M <_ 2 , M , 2 ) ) ) -> (π ` M ) = ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) )
48	2, 12, 47	syl2anc 693	. . 3 |- ( N e. ( ZZ>= ` M ) -> (π ` M ) = ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) )
49	46, 48	oveq12d 6668	. 2 |- ( N e. ( ZZ>= ` M ) -> ( (π ` N ) - (π ` M ) ) = ( ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) - ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) ) )
50		hashcl 13147	. . . . 5 |- ( ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) e. Fin -> ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) e. NN0 )
51	38, 50	ax-mp 5	. . . 4 |- ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) e. NN0
52	51	nn0cni 11304	. . 3 |- ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) e. CC
53		hashcl 13147	. . . . 5 |- ( ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin -> ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. NN0 )
54	42, 53	ax-mp 5	. . . 4 |- ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. NN0
55	54	nn0cni 11304	. . 3 |- ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. CC
56		pncan2 10288	. . 3 |- ( ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) e. CC /\ ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. CC ) -> ( ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) - ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) ) = ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) )
57	52, 55, 56	mp2an 708	. 2 |- ( ( ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) + ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) - ( # ` ( ( if ( M <_ 2 , M , 2 ) ... M ) i^i Prime ) ) ) = ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) )
58	49, 57	syl6eq 2672	1 |- ( N e. ( ZZ>= ` M ) -> ( (π ` N ) - (π ` M ) ) = ( # ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117   Primecprime 15385  πcppi 24820

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-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-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-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-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-2o 7561  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-card 8765  df-cda 8990  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-icc 12182  df-fz 12327  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-dvds 14984  df-prm 15386  df-ppi 24826

This theorem is referenced by:  ppiub  24929  chtppilimlem1  25162