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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz2 | Structured version Visualization version GIF version |
Description: Special case of hashnzfz 39485: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
hashnzfz2.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) |
hashnzfz2.k | ⊢ (𝜑 → 𝐾 ∈ ℕ) |
Ref | Expression |
---|---|
hashnzfz2 | ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 11452 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | uznnssnn 12045 | . . . . 5 ⊢ (2 ∈ ℕ → (ℤ≥‘2) ⊆ ℕ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (ℤ≥‘2) ⊆ ℕ |
4 | hashnzfz2.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) | |
5 | 3, 4 | sseldi 3819 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 2z 11765 | . . . 4 ⊢ 2 ∈ ℤ | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ∈ ℤ) |
8 | hashnzfz2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
9 | nnuz 12033 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 2m1e1 11512 | . . . . . 6 ⊢ (2 − 1) = 1 | |
11 | 10 | fveq2i 6451 | . . . . 5 ⊢ (ℤ≥‘(2 − 1)) = (ℤ≥‘1) |
12 | 9, 11 | eqtr4i 2805 | . . . 4 ⊢ ℕ = (ℤ≥‘(2 − 1)) |
13 | 8, 12 | syl6eleq 2869 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(2 − 1))) |
14 | 5, 7, 13 | hashnzfz 39485 | . 2 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((2 − 1) / 𝑁)))) |
15 | 10 | oveq1i 6934 | . . . . 5 ⊢ ((2 − 1) / 𝑁) = (1 / 𝑁) |
16 | 15 | fveq2i 6451 | . . . 4 ⊢ (⌊‘((2 − 1) / 𝑁)) = (⌊‘(1 / 𝑁)) |
17 | 0red 10382 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 5 | nnrecred 11430 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
19 | 5 | nnred 11395 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | 5 | nngt0d 11428 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑁) |
21 | 19, 20 | recgt0d 11314 | . . . . . 6 ⊢ (𝜑 → 0 < (1 / 𝑁)) |
22 | 17, 18, 21 | ltled 10526 | . . . . 5 ⊢ (𝜑 → 0 ≤ (1 / 𝑁)) |
23 | eluzle 12009 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → 2 ≤ 𝑁) | |
24 | 4, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≤ 𝑁) |
25 | 5 | nnzd 11837 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
26 | zlem1lt 11785 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 − 1) < 𝑁)) | |
27 | 6, 25, 26 | sylancr 581 | . . . . . . . . 9 ⊢ (𝜑 → (2 ≤ 𝑁 ↔ (2 − 1) < 𝑁)) |
28 | 24, 27 | mpbid 224 | . . . . . . . 8 ⊢ (𝜑 → (2 − 1) < 𝑁) |
29 | 10, 28 | syl5eqbrr 4924 | . . . . . . 7 ⊢ (𝜑 → 1 < 𝑁) |
30 | 5 | nnrpd 12183 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ+) |
31 | 30 | recgt1d 12199 | . . . . . . 7 ⊢ (𝜑 → (1 < 𝑁 ↔ (1 / 𝑁) < 1)) |
32 | 29, 31 | mpbid 224 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑁) < 1) |
33 | 0p1e1 11508 | . . . . . 6 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | syl6breqr 4930 | . . . . 5 ⊢ (𝜑 → (1 / 𝑁) < (0 + 1)) |
35 | 0z 11743 | . . . . . 6 ⊢ 0 ∈ ℤ | |
36 | flbi 12940 | . . . . . 6 ⊢ (((1 / 𝑁) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘(1 / 𝑁)) = 0 ↔ (0 ≤ (1 / 𝑁) ∧ (1 / 𝑁) < (0 + 1)))) | |
37 | 18, 35, 36 | sylancl 580 | . . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝑁)) = 0 ↔ (0 ≤ (1 / 𝑁) ∧ (1 / 𝑁) < (0 + 1)))) |
38 | 22, 34, 37 | mpbir2and 703 | . . . 4 ⊢ (𝜑 → (⌊‘(1 / 𝑁)) = 0) |
39 | 16, 38 | syl5eq 2826 | . . 3 ⊢ (𝜑 → (⌊‘((2 − 1) / 𝑁)) = 0) |
40 | 39 | oveq2d 6940 | . 2 ⊢ (𝜑 → ((⌊‘(𝐾 / 𝑁)) − (⌊‘((2 − 1) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − 0)) |
41 | 8 | nnred 11395 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
42 | 41, 5 | nndivred 11433 | . . . . 5 ⊢ (𝜑 → (𝐾 / 𝑁) ∈ ℝ) |
43 | 42 | flcld 12922 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐾 / 𝑁)) ∈ ℤ) |
44 | 43 | zcnd 11839 | . . 3 ⊢ (𝜑 → (⌊‘(𝐾 / 𝑁)) ∈ ℂ) |
45 | 44 | subid1d 10725 | . 2 ⊢ (𝜑 → ((⌊‘(𝐾 / 𝑁)) − 0) = (⌊‘(𝐾 / 𝑁))) |
46 | 14, 40, 45 | 3eqtrd 2818 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 {csn 4398 class class class wbr 4888 “ cima 5360 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 < clt 10413 ≤ cle 10414 − cmin 10608 / cdiv 11034 ℕcn 11378 2c2 11434 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-2 11442 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-fz 12648 df-fl 12916 df-hash 13440 df-dvds 15392 |
This theorem is referenced by: (None) |
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