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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz | Structured version Visualization version GIF version |
Description: Special case of hashdvds 16100: 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 11981 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
5 | 1, 2, 3, 4 | hashdvds 16100 | . 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)}) = ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁)))) |
6 | elfzelz 12896 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℤ) | |
7 | 6 | zcnd 12076 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℂ) |
8 | 7 | subid1d 10974 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑥 − 0) = 𝑥) |
9 | 8 | breq2d 5069 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑁 ∥ (𝑥 − 0) ↔ 𝑁 ∥ 𝑥)) |
10 | 9 | rabbiia 3470 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} |
11 | dfrab3 4275 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
12 | reldvds 40524 | . . . . . . . 8 ⊢ Rel ∥ | |
13 | relimasn 5945 | . . . . . . . 8 ⊢ (Rel ∥ → ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥} |
15 | 14 | ineq2i 4183 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) |
16 | incom 4175 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) | |
17 | 15, 16 | eqtr3i 2843 | . . . . 5 ⊢ ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
18 | 10, 11, 17 | 3eqtri 2845 | . . . 4 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
19 | 18 | fveq2i 6666 | . . 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 12241 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝐽 − 1)) → 𝐾 ∈ ℤ) | |
22 | 3, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
23 | 22 | zcnd 12076 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
24 | 23 | subid1d 10974 | . . . 4 ⊢ (𝜑 → (𝐾 − 0) = 𝐾) |
25 | 24 | fvoveq1d 7167 | . . 3 ⊢ (𝜑 → (⌊‘((𝐾 − 0) / 𝑁)) = (⌊‘(𝐾 / 𝑁))) |
26 | peano2zm 12013 | . . . . . . 7 ⊢ (𝐽 ∈ ℤ → (𝐽 − 1) ∈ ℤ) | |
27 | 2, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ) |
28 | 27 | zcnd 12076 | . . . . 5 ⊢ (𝜑 → (𝐽 − 1) ∈ ℂ) |
29 | 28 | subid1d 10974 | . . . 4 ⊢ (𝜑 → ((𝐽 − 1) − 0) = (𝐽 − 1)) |
30 | 29 | fvoveq1d 7167 | . . 3 ⊢ (𝜑 → (⌊‘(((𝐽 − 1) − 0) / 𝑁)) = (⌊‘((𝐽 − 1) / 𝑁))) |
31 | 25, 30 | oveq12d 7163 | . 2 ⊢ (𝜑 → ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
32 | 5, 20, 31 | 3eqtr3d 2861 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {cab 2796 {crab 3139 ∩ cin 3932 {csn 4557 class class class wbr 5057 “ cima 5551 Rel wrel 5553 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 − cmin 10858 / cdiv 11285 ℕcn 11626 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 ⌊cfl 13148 ♯chash 13678 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fl 13150 df-hash 13679 df-dvds 15596 |
This theorem is referenced by: hashnzfz2 40530 hashnzfzclim 40531 |
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