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Mirrors > Home > MPE Home > Th. List > nndivdvds | Structured version Visualization version GIF version |
Description: Strong form of dvdsval2 16259 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 12631 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
2 | nnne0 12298 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
3 | nnz 12631 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
4 | 3 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ) |
5 | dvdsval2 16259 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℤ)) | |
6 | 1, 2, 4, 5 | syl2an23an 1420 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℤ)) |
7 | 6 | anbi1d 629 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)) ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
8 | nnre 12271 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
9 | 8 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | nnre 12271 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
11 | 10 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
12 | nngt0 12295 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
13 | 12 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐴) |
14 | nngt0 12295 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
15 | 14 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
16 | 9, 11, 13, 15 | divgt0d 12201 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < (𝐴 / 𝐵)) |
17 | 16 | biantrud 530 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)))) |
18 | elnnz 12620 | . . 3 ⊢ ((𝐴 / 𝐵) ∈ ℕ ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵))) | |
19 | 18 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
20 | 7, 17, 19 | 3bitr4d 310 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 (class class class)co 7424 ℝcr 11157 0cc0 11158 < clt 11298 / cdiv 11921 ℕcn 12264 ℤcz 12610 ∥ cdvds 16256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-z 12611 df-dvds 16257 |
This theorem is referenced by: nndivides 16266 dvdsdivcl 16318 divgcdnn 16515 lcmgcdlem 16607 isprm6 16715 divnumden 16750 hashgcdlem 16790 hashgcdeq 16791 oddprmdvds 16905 gexexlem 19850 ablfac1lem 20068 pgpfac1lem3a 20076 fincygsubgodexd 20113 znrrg 21563 dvdsflf1o 27215 mersenne 27256 perfectlem1 27258 perfect 27260 dchrvmasumlem1 27524 dchrisum0flblem2 27538 logsqvma 27571 oddpwdc 34188 nndivdvdsd 41698 lcmineqlem4 41731 lcmineqlem23 41750 aks6d1c1p3 41808 aks6d1c2p1 41816 aks6d1c2p2 41817 dffltz 42288 jm2.20nn 42655 jm2.27c 42665 fouriersw 45852 proththdlem 47185 perfectALTVlem1 47293 perfectALTV 47295 |
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