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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz | Structured version Visualization version GIF version |
Description: Special case of hashdvds 16487: 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 12342 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
5 | 1, 2, 3, 4 | hashdvds 16487 | . 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)}) = ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁)))) |
6 | elfzelz 13267 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℤ) | |
7 | 6 | zcnd 12438 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℂ) |
8 | 7 | subid1d 11332 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑥 − 0) = 𝑥) |
9 | 8 | breq2d 5091 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑁 ∥ (𝑥 − 0) ↔ 𝑁 ∥ 𝑥)) |
10 | 9 | rabbiia 3405 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} |
11 | dfrab3 4249 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
12 | reldvds 41915 | . . . . . . . 8 ⊢ Rel ∥ | |
13 | relimasn 5991 | . . . . . . . 8 ⊢ (Rel ∥ → ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥} |
15 | 14 | ineq2i 4149 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) |
16 | incom 4140 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) | |
17 | 15, 16 | eqtr3i 2770 | . . . . 5 ⊢ ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
18 | 10, 11, 17 | 3eqtri 2772 | . . . 4 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
19 | 18 | fveq2i 6774 | . . 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 12603 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝐽 − 1)) → 𝐾 ∈ ℤ) | |
22 | 3, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
23 | 22 | zcnd 12438 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
24 | 23 | subid1d 11332 | . . . 4 ⊢ (𝜑 → (𝐾 − 0) = 𝐾) |
25 | 24 | fvoveq1d 7294 | . . 3 ⊢ (𝜑 → (⌊‘((𝐾 − 0) / 𝑁)) = (⌊‘(𝐾 / 𝑁))) |
26 | peano2zm 12374 | . . . . . . 7 ⊢ (𝐽 ∈ ℤ → (𝐽 − 1) ∈ ℤ) | |
27 | 2, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ) |
28 | 27 | zcnd 12438 | . . . . 5 ⊢ (𝜑 → (𝐽 − 1) ∈ ℂ) |
29 | 28 | subid1d 11332 | . . . 4 ⊢ (𝜑 → ((𝐽 − 1) − 0) = (𝐽 − 1)) |
30 | 29 | fvoveq1d 7294 | . . 3 ⊢ (𝜑 → (⌊‘(((𝐽 − 1) − 0) / 𝑁)) = (⌊‘((𝐽 − 1) / 𝑁))) |
31 | 25, 30 | oveq12d 7290 | . 2 ⊢ (𝜑 → ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
32 | 5, 20, 31 | 3eqtr3d 2788 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {cab 2717 {crab 3070 ∩ cin 3891 {csn 4567 class class class wbr 5079 “ cima 5593 Rel wrel 5595 ‘cfv 6432 (class class class)co 7272 0cc0 10882 1c1 10883 − cmin 11216 / cdiv 11643 ℕcn 11984 ℤcz 12330 ℤ≥cuz 12593 ...cfz 13250 ⌊cfl 13521 ♯chash 14055 ∥ cdvds 15974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-sup 9189 df-inf 9190 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-n0 12245 df-z 12331 df-uz 12594 df-fz 13251 df-fl 13523 df-hash 14056 df-dvds 15975 |
This theorem is referenced by: hashnzfz2 41921 hashnzfzclim 41922 |
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