Mathbox for Steve Rodriguez |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz2 | Structured version Visualization version GIF version |
Description: Special case of hashnzfz 40659: 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 11713 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | uznnssnn 12298 | . . . . 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 3967 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 2z 12017 | . . . 4 ⊢ 2 ∈ ℤ | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ∈ ℤ) |
8 | hashnzfz2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
9 | nnuz 12284 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 2m1e1 11766 | . . . . . 6 ⊢ (2 − 1) = 1 | |
11 | 10 | fveq2i 6675 | . . . . 5 ⊢ (ℤ≥‘(2 − 1)) = (ℤ≥‘1) |
12 | 9, 11 | eqtr4i 2849 | . . . 4 ⊢ ℕ = (ℤ≥‘(2 − 1)) |
13 | 8, 12 | eleqtrdi 2925 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(2 − 1))) |
14 | 5, 7, 13 | hashnzfz 40659 | . 2 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((2 − 1) / 𝑁)))) |
15 | 10 | oveq1i 7168 | . . . . 5 ⊢ ((2 − 1) / 𝑁) = (1 / 𝑁) |
16 | 15 | fveq2i 6675 | . . . 4 ⊢ (⌊‘((2 − 1) / 𝑁)) = (⌊‘(1 / 𝑁)) |
17 | 0red 10646 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 5 | nnrecred 11691 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
19 | 5 | nnred 11655 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | 5 | nngt0d 11689 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑁) |
21 | 19, 20 | recgt0d 11576 | . . . . . 6 ⊢ (𝜑 → 0 < (1 / 𝑁)) |
22 | 17, 18, 21 | ltled 10790 | . . . . 5 ⊢ (𝜑 → 0 ≤ (1 / 𝑁)) |
23 | eluzle 12259 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → 2 ≤ 𝑁) | |
24 | 4, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≤ 𝑁) |
25 | 5 | nnzd 12089 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
26 | zlem1lt 12037 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 − 1) < 𝑁)) | |
27 | 6, 25, 26 | sylancr 589 | . . . . . . . . 9 ⊢ (𝜑 → (2 ≤ 𝑁 ↔ (2 − 1) < 𝑁)) |
28 | 24, 27 | mpbid 234 | . . . . . . . 8 ⊢ (𝜑 → (2 − 1) < 𝑁) |
29 | 10, 28 | eqbrtrrid 5104 | . . . . . . 7 ⊢ (𝜑 → 1 < 𝑁) |
30 | 5 | nnrpd 12432 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ+) |
31 | 30 | recgt1d 12448 | . . . . . . 7 ⊢ (𝜑 → (1 < 𝑁 ↔ (1 / 𝑁) < 1)) |
32 | 29, 31 | mpbid 234 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑁) < 1) |
33 | 0p1e1 11762 | . . . . . 6 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | breqtrrdi 5110 | . . . . 5 ⊢ (𝜑 → (1 / 𝑁) < (0 + 1)) |
35 | 0z 11995 | . . . . . 6 ⊢ 0 ∈ ℤ | |
36 | flbi 13189 | . . . . . 6 ⊢ (((1 / 𝑁) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘(1 / 𝑁)) = 0 ↔ (0 ≤ (1 / 𝑁) ∧ (1 / 𝑁) < (0 + 1)))) | |
37 | 18, 35, 36 | sylancl 588 | . . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝑁)) = 0 ↔ (0 ≤ (1 / 𝑁) ∧ (1 / 𝑁) < (0 + 1)))) |
38 | 22, 34, 37 | mpbir2and 711 | . . . 4 ⊢ (𝜑 → (⌊‘(1 / 𝑁)) = 0) |
39 | 16, 38 | syl5eq 2870 | . . 3 ⊢ (𝜑 → (⌊‘((2 − 1) / 𝑁)) = 0) |
40 | 39 | oveq2d 7174 | . 2 ⊢ (𝜑 → ((⌊‘(𝐾 / 𝑁)) − (⌊‘((2 − 1) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − 0)) |
41 | 8 | nnred 11655 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
42 | 41, 5 | nndivred 11694 | . . . . 5 ⊢ (𝜑 → (𝐾 / 𝑁) ∈ ℝ) |
43 | 42 | flcld 13171 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐾 / 𝑁)) ∈ ℤ) |
44 | 43 | zcnd 12091 | . . 3 ⊢ (𝜑 → (⌊‘(𝐾 / 𝑁)) ∈ ℂ) |
45 | 44 | subid1d 10988 | . 2 ⊢ (𝜑 → ((⌊‘(𝐾 / 𝑁)) − 0) = (⌊‘(𝐾 / 𝑁))) |
46 | 14, 40, 45 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 {csn 4569 class class class wbr 5068 “ cima 5560 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 ℤcz 11984 ℤ≥cuz 12246 ...cfz 12895 ⌊cfl 13163 ♯chash 13693 ∥ cdvds 15609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fl 13165 df-hash 13694 df-dvds 15610 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |