Theorem hashnzfz 38519

Description: Special case of hashdvds 15480: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Hypotheses

Ref	Expression
hashnzfz.n	|- ( ph -> N e. NN )
hashnzfz.j	|- ( ph -> J e. ZZ )
hashnzfz.k	|- ( ph -> K e. ( ZZ>= ` ( J - 1 ) ) )

Assertion

Ref	Expression
hashnzfz	|- ( ph -> ( # ` ( ( || " { N } ) i^i ( J ... K ) ) ) = ( ( |_ ` ( K / N ) ) - ( |_ ` ( ( J - 1 ) / N ) ) ) )

Proof of Theorem hashnzfz

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashnzfz.n	. . 3 |- ( ph -> N e. NN )
2		hashnzfz.j	. . 3 |- ( ph -> J e. ZZ )
3		hashnzfz.k	. . 3 |- ( ph -> K e. ( ZZ>= ` ( J - 1 ) ) )
4		0zd 11389	. . 3 |- ( ph -> 0 e. ZZ )
5	1, 2, 3, 4	hashdvds 15480	. 2 |- ( ph -> ( # ` { x e. ( J ... K ) | N || ( x - 0 ) } ) = ( ( |_ ` ( ( K - 0 ) / N ) ) - ( |_ ` ( ( ( J - 1 ) - 0 ) / N ) ) ) )
6		elfzelz 12342	. . . . . . . . 9 |- ( x e. ( J ... K ) -> x e. ZZ )
7	6	zcnd 11483	. . . . . . . 8 |- ( x e. ( J ... K ) -> x e. CC )
8	7	subid1d 10381	. . . . . . 7 |- ( x e. ( J ... K ) -> ( x - 0 ) = x )
9	8	breq2d 4665	. . . . . 6 |- ( x e. ( J ... K ) -> ( N || ( x - 0 ) <-> N || x ) )
10	9	rabbiia 3185	. . . . 5 |- { x e. ( J ... K ) | N || ( x - 0 ) } = { x e. ( J ... K ) | N || x }
11		dfrab3 3902	. . . . 5 |- { x e. ( J ... K ) | N || x } = ( ( J ... K ) i^i { x | N || x } )
12		reldvds 38514	. . . . . . . 8 |- Rel ||
13		relimasn 5488	. . . . . . . 8 |- ( Rel || -> ( || " { N } ) = { x | N || x } )
14	12, 13	ax-mp 5	. . . . . . 7 |- ( || " { N } ) = { x | N || x }
15	14	ineq2i 3811	. . . . . 6 |- ( ( J ... K ) i^i ( || " { N } ) ) = ( ( J ... K ) i^i { x | N || x } )
16		incom 3805	. . . . . 6 |- ( ( J ... K ) i^i ( || " { N } ) ) = ( ( || " { N } ) i^i ( J ... K ) )
17	15, 16	eqtr3i 2646	. . . . 5 |- ( ( J ... K ) i^i { x | N || x } ) = ( ( || " { N } ) i^i ( J ... K ) )
18	10, 11, 17	3eqtri 2648	. . . 4 |- { x e. ( J ... K ) | N || ( x - 0 ) } = ( ( || " { N } ) i^i ( J ... K ) )
19	18	fveq2i 6194	. . 3 |- ( # ` { x e. ( J ... K ) | N || ( x - 0 ) } ) = ( # ` ( ( || " { N } ) i^i ( J ... K ) ) )
20	19	a1i 11	. 2 |- ( ph -> ( # ` { x e. ( J ... K ) | N || ( x - 0 ) } ) = ( # ` ( ( || " { N } ) i^i ( J ... K ) ) ) )
21		eluzelz 11697	. . . . . . . 8 |- ( K e. ( ZZ>= ` ( J - 1 ) ) -> K e. ZZ )
22	3, 21	syl 17	. . . . . . 7 |- ( ph -> K e. ZZ )
23	22	zcnd 11483	. . . . . 6 |- ( ph -> K e. CC )
24	23	subid1d 10381	. . . . 5 |- ( ph -> ( K - 0 ) = K )
25	24	oveq1d 6665	. . . 4 |- ( ph -> ( ( K - 0 ) / N ) = ( K / N ) )
26	25	fveq2d 6195	. . 3 |- ( ph -> ( |_ ` ( ( K - 0 ) / N ) ) = ( |_ ` ( K / N ) ) )
27		peano2zm 11420	. . . . . . . 8 |- ( J e. ZZ -> ( J - 1 ) e. ZZ )
28	2, 27	syl 17	. . . . . . 7 |- ( ph -> ( J - 1 ) e. ZZ )
29	28	zcnd 11483	. . . . . 6 |- ( ph -> ( J - 1 ) e. CC )
30	29	subid1d 10381	. . . . 5 |- ( ph -> ( ( J - 1 ) - 0 ) = ( J - 1 ) )
31	30	oveq1d 6665	. . . 4 |- ( ph -> ( ( ( J - 1 ) - 0 ) / N ) = ( ( J - 1 ) / N ) )
32	31	fveq2d 6195	. . 3 |- ( ph -> ( |_ ` ( ( ( J - 1 ) - 0 ) / N ) ) = ( |_ ` ( ( J - 1 ) / N ) ) )
33	26, 32	oveq12d 6668	. 2 |- ( ph -> ( ( |_ ` ( ( K - 0 ) / N ) ) - ( |_ ` ( ( ( J - 1 ) - 0 ) / N ) ) ) = ( ( |_ ` ( K / N ) ) - ( |_ ` ( ( J - 1 ) / N ) ) ) )
34	5, 20, 33	3eqtr3d 2664	1 |- ( ph -> ( # ` ( ( || " { N } ) i^i ( J ... K ) ) ) = ( ( |_ ` ( K / N ) ) - ( |_ ` ( ( J - 1 ) / N ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   i^i cin 3573   {csn 4177   class class class wbr 4653   "cima 5117   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   #chash 13117   || 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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fl 12593  df-hash 13118  df-dvds 14984

This theorem is referenced by:  hashnzfz2  38520  hashnzfzclim  38521