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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashnzfz2 | Structured version Visualization version GIF version |
Description: Special case of hashnzfz 44316: 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 12337 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | uznnssnn 12935 | . . . . 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 | sselid 3993 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 2z 12647 | . . . 4 ⊢ 2 ∈ ℤ | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ∈ ℤ) |
8 | hashnzfz2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
9 | nnuz 12919 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 2m1e1 12390 | . . . . . 6 ⊢ (2 − 1) = 1 | |
11 | 10 | fveq2i 6910 | . . . . 5 ⊢ (ℤ≥‘(2 − 1)) = (ℤ≥‘1) |
12 | 9, 11 | eqtr4i 2766 | . . . 4 ⊢ ℕ = (ℤ≥‘(2 − 1)) |
13 | 8, 12 | eleqtrdi 2849 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(2 − 1))) |
14 | 5, 7, 13 | hashnzfz 44316 | . 2 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((2 − 1) / 𝑁)))) |
15 | 10 | oveq1i 7441 | . . . . 5 ⊢ ((2 − 1) / 𝑁) = (1 / 𝑁) |
16 | 15 | fveq2i 6910 | . . . 4 ⊢ (⌊‘((2 − 1) / 𝑁)) = (⌊‘(1 / 𝑁)) |
17 | 0red 11262 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 5 | nnrecred 12315 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
19 | 5 | nnred 12279 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | 5 | nngt0d 12313 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑁) |
21 | 19, 20 | recgt0d 12200 | . . . . . 6 ⊢ (𝜑 → 0 < (1 / 𝑁)) |
22 | 17, 18, 21 | ltled 11407 | . . . . 5 ⊢ (𝜑 → 0 ≤ (1 / 𝑁)) |
23 | eluzle 12889 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → 2 ≤ 𝑁) | |
24 | 4, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≤ 𝑁) |
25 | 5 | nnzd 12638 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
26 | zlem1lt 12667 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 − 1) < 𝑁)) | |
27 | 6, 25, 26 | sylancr 587 | . . . . . . . . 9 ⊢ (𝜑 → (2 ≤ 𝑁 ↔ (2 − 1) < 𝑁)) |
28 | 24, 27 | mpbid 232 | . . . . . . . 8 ⊢ (𝜑 → (2 − 1) < 𝑁) |
29 | 10, 28 | eqbrtrrid 5184 | . . . . . . 7 ⊢ (𝜑 → 1 < 𝑁) |
30 | 5 | nnrpd 13073 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ+) |
31 | 30 | recgt1d 13089 | . . . . . . 7 ⊢ (𝜑 → (1 < 𝑁 ↔ (1 / 𝑁) < 1)) |
32 | 29, 31 | mpbid 232 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑁) < 1) |
33 | 0p1e1 12386 | . . . . . 6 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | breqtrrdi 5190 | . . . . 5 ⊢ (𝜑 → (1 / 𝑁) < (0 + 1)) |
35 | 0z 12622 | . . . . . 6 ⊢ 0 ∈ ℤ | |
36 | flbi 13853 | . . . . . 6 ⊢ (((1 / 𝑁) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘(1 / 𝑁)) = 0 ↔ (0 ≤ (1 / 𝑁) ∧ (1 / 𝑁) < (0 + 1)))) | |
37 | 18, 35, 36 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝑁)) = 0 ↔ (0 ≤ (1 / 𝑁) ∧ (1 / 𝑁) < (0 + 1)))) |
38 | 22, 34, 37 | mpbir2and 713 | . . . 4 ⊢ (𝜑 → (⌊‘(1 / 𝑁)) = 0) |
39 | 16, 38 | eqtrid 2787 | . . 3 ⊢ (𝜑 → (⌊‘((2 − 1) / 𝑁)) = 0) |
40 | 39 | oveq2d 7447 | . 2 ⊢ (𝜑 → ((⌊‘(𝐾 / 𝑁)) − (⌊‘((2 − 1) / 𝑁))) = ((⌊‘(𝐾 / 𝑁)) − 0)) |
41 | 8 | nnred 12279 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
42 | 41, 5 | nndivred 12318 | . . . . 5 ⊢ (𝜑 → (𝐾 / 𝑁) ∈ ℝ) |
43 | 42 | flcld 13835 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐾 / 𝑁)) ∈ ℤ) |
44 | 43 | zcnd 12721 | . . 3 ⊢ (𝜑 → (⌊‘(𝐾 / 𝑁)) ∈ ℂ) |
45 | 44 | subid1d 11607 | . 2 ⊢ (𝜑 → ((⌊‘(𝐾 / 𝑁)) − 0) = (⌊‘(𝐾 / 𝑁))) |
46 | 14, 40, 45 | 3eqtrd 2779 | 1 ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 {csn 4631 class class class wbr 5148 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 ⌊cfl 13827 ♯chash 14366 ∥ cdvds 16287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fl 13829 df-hash 14367 df-dvds 16288 |
This theorem is referenced by: (None) |
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