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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz | Structured version Visualization version GIF version |
Description: Special case of hashdvds 15888: 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 11744 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
5 | 1, 2, 3, 4 | hashdvds 15888 | . 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)}) = ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁)))) |
6 | elfzelz 12663 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℤ) | |
7 | 6 | zcnd 11839 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ∈ ℂ) |
8 | 7 | subid1d 10725 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑥 − 0) = 𝑥) |
9 | 8 | breq2d 4900 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽...𝐾) → (𝑁 ∥ (𝑥 − 0) ↔ 𝑁 ∥ 𝑥)) |
10 | 9 | rabbiia 3381 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} |
11 | dfrab3 4128 | . . . . 5 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ 𝑥} = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
12 | reldvds 39480 | . . . . . . . 8 ⊢ Rel ∥ | |
13 | relimasn 5744 | . . . . . . . 8 ⊢ (Rel ∥ → ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥}) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥} |
15 | 14 | ineq2i 4034 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) |
16 | incom 4028 | . . . . . 6 ⊢ ((𝐽...𝐾) ∩ ( ∥ “ {𝑁})) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) | |
17 | 15, 16 | eqtr3i 2804 | . . . . 5 ⊢ ((𝐽...𝐾) ∩ {𝑥 ∣ 𝑁 ∥ 𝑥}) = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
18 | 10, 11, 17 | 3eqtri 2806 | . . . 4 ⊢ {𝑥 ∈ (𝐽...𝐾) ∣ 𝑁 ∥ (𝑥 − 0)} = (( ∥ “ {𝑁}) ∩ (𝐽...𝐾)) |
19 | 18 | fveq2i 6451 | . . 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 12006 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝐽 − 1)) → 𝐾 ∈ ℤ) | |
22 | 3, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
23 | 22 | zcnd 11839 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
24 | 23 | subid1d 10725 | . . . 4 ⊢ (𝜑 → (𝐾 − 0) = 𝐾) |
25 | 24 | fvoveq1d 6946 | . . 3 ⊢ (𝜑 → (⌊‘((𝐾 − 0) / 𝑁)) = (⌊‘(𝐾 / 𝑁))) |
26 | peano2zm 11776 | . . . . . . 7 ⊢ (𝐽 ∈ ℤ → (𝐽 − 1) ∈ ℤ) | |
27 | 2, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ) |
28 | 27 | zcnd 11839 | . . . . 5 ⊢ (𝜑 → (𝐽 − 1) ∈ ℂ) |
29 | 28 | subid1d 10725 | . . . 4 ⊢ (𝜑 → ((𝐽 − 1) − 0) = (𝐽 − 1)) |
30 | 29 | fvoveq1d 6946 | . . 3 ⊢ (𝜑 → (⌊‘(((𝐽 − 1) − 0) / 𝑁)) = (⌊‘((𝐽 − 1) / 𝑁))) |
31 | 25, 30 | oveq12d 6942 | . 2 ⊢ (𝜑 → ((⌊‘((𝐾 − 0) / 𝑁)) − (⌊‘(((𝐽 − 1) − 0) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
32 | 5, 20, 31 | 3eqtr3d 2822 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {cab 2763 {crab 3094 ∩ cin 3791 {csn 4398 class class class wbr 4888 “ cima 5360 Rel wrel 5362 ‘cfv 6137 (class class class)co 6924 0cc0 10274 1c1 10275 − cmin 10608 / cdiv 11034 ℕcn 11378 ℤcz 11732 ℤ≥cuz 11996 ...cfz 12647 ⌊cfl 12914 ♯chash 13439 ∥ cdvds 15391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-fl 12916 df-hash 13440 df-dvds 15392 |
This theorem is referenced by: hashnzfz2 39486 hashnzfzclim 39487 |
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