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Mirrors > Home > MPE Home > Th. List > nndivdvds | Structured version Visualization version GIF version |
Description: Strong form of dvdsval2 15360 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
nndivdvds | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 11727 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
2 | 1 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ) |
3 | nnne0 11386 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
4 | 3 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0) |
5 | nnz 11727 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
6 | 5 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ) |
7 | dvdsval2 15360 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℤ)) | |
8 | 2, 4, 6, 7 | syl3anc 1494 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℤ)) |
9 | 8 | anbi1d 623 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)) ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
10 | nnre 11358 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
11 | 10 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ) |
12 | nnre 11358 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
13 | 12 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
14 | nngt0 11383 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
15 | 14 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐴) |
16 | nngt0 11383 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
17 | 16 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
18 | 11, 13, 15, 17 | divgt0d 11289 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < (𝐴 / 𝐵)) |
19 | 18 | biantrud 527 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)))) |
20 | elnnz 11714 | . . 3 ⊢ ((𝐴 / 𝐵) ∈ ℕ ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵))) | |
21 | 20 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
22 | 9, 19, 21 | 3bitr4d 303 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 0cc0 10252 < clt 10391 / cdiv 11009 ℕcn 11350 ℤcz 11704 ∥ cdvds 15357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-z 11705 df-dvds 15358 |
This theorem is referenced by: nndivides 15367 dvdsdivcl 15415 divgcdnn 15609 lcmgcdlem 15692 isprm6 15797 divnumden 15827 hashgcdlem 15864 hashgcdeq 15865 oddprmdvds 15978 gexexlem 18608 ablfac1lem 18821 pgpfac1lem3a 18829 znrrg 20273 dvdsflf1o 25326 mersenne 25365 perfectlem1 25367 perfect 25369 dchrvmasumlem1 25597 dchrisum0flblem2 25611 logsqvma 25644 oddpwdc 30950 dffltz 38090 jm2.20nn 38400 jm2.27c 38410 fouriersw 41235 proththdlem 42353 perfectALTVlem1 42453 perfectALTV 42455 |
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