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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz | Structured version Visualization version GIF version |
Description: Special case of hashdvds 16504: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
hashnzfz.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
hashnzfz.j | ⊢ (𝜑 → 𝐽 ∈ ℤ) |
hashnzfz.k | ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(𝐽 − 1))) |
Ref | Expression |
---|---|
hashnzfz | ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnzfz.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
2 | hashnzfz.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ ℤ) | |
3 | hashnzfz.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(𝐽 − 1))) | |
4 | 0zd 12359 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
5 | 1, 2, 3, 4 | hashdvds 16504 | . 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)}) = ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁)))) |
6 | elfzelz 13284 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℤ) | |
7 | 6 | zcnd 12455 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℂ) |
8 | 7 | subid1d 11349 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑥 − 0) = 𝑥) |
9 | 8 | breq2d 5089 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑁 ∥ (𝑥 − 0) ↔ 𝑁 ∥ 𝑥)) |
10 | 9 | rabbiia 3409 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} |
11 | dfrab3 4246 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
12 | reldvds 41957 | . . . . . . . 8 ⊢ Rel ∥ | |
13 | relimasn 5993 | . . . . . . . 8 ⊢ (Rel ∥ → ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥} |
15 | 14 | ineq2i 4146 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) |
16 | incom 4138 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) | |
17 | 15, 16 | eqtr3i 2763 | . . . . 5 ⊢ ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
18 | 10, 11, 17 | 3eqtri 2765 | . . . 4 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
19 | 18 | fveq2i 6795 | . . 3 ⊢ (♯‘{𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)}) = (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)}) = (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾)))) |
21 | eluzelz 12620 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝐽 − 1)) → 𝐾 ∈ ℤ) | |
22 | 3, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
23 | 22 | zcnd 12455 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
24 | 23 | subid1d 11349 | . . . 4 ⊢ (𝜑 → (𝐾 − 0) = 𝐾) |
25 | 24 | fvoveq1d 7317 | . . 3 ⊢ (𝜑 → (⌊‘((𝐾 − 0) / 𝑁)) = (⌊‘(𝐾 / 𝑁))) |
26 | peano2zm 12391 | . . . . . . 7 ⊢ (𝐽 ∈ ℤ → (𝐽 − 1) ∈ ℤ) | |
27 | 2, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ) |
28 | 27 | zcnd 12455 | . . . . 5 ⊢ (𝜑 → (𝐽 − 1) ∈ ℂ) |
29 | 28 | subid1d 11349 | . . . 4 ⊢ (𝜑 → ((𝐽 − 1) − 0) = (𝐽 − 1)) |
30 | 29 | fvoveq1d 7317 | . . 3 ⊢ (𝜑 → (⌊‘(((𝐽 − 1) − 0) / 𝑁)) = (⌊‘((𝐽 − 1) / 𝑁))) |
31 | 25, 30 | oveq12d 7313 | . 2 ⊢ (𝜑 → ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
32 | 5, 20, 31 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 {cab 2710 {crab 3221 ∩ cin 3888 {csn 4564 class class class wbr 5077 “ cima 5594 Rel wrel 5596 ‘cfv 6447 (class class class)co 7295 0cc0 10899 1c1 10900 − cmin 11233 / cdiv 11660 ℕcn 12001 ℤcz 12347 ℤ≥cuz 12610 ...cfz 13267 ⌊cfl 13538 ♯chash 14072 ∥ cdvds 15991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-inf 9230 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-n0 12262 df-z 12348 df-uz 12611 df-fz 13268 df-fl 13540 df-hash 14073 df-dvds 15992 |
This theorem is referenced by: hashnzfz2 41963 hashnzfzclim 41964 |
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