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Mirrors > Home > MPE Home > Th. List > nndivdvds | Structured version Visualization version GIF version |
Description: Strong form of dvdsval2 15602 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 11992 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
2 | nnne0 11659 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
3 | nnz 11992 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ) |
5 | dvdsval2 15602 | . . . 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 632 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)) ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
8 | nnre 11632 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
9 | 8 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | nnre 11632 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
11 | 10 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
12 | nngt0 11656 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
13 | 12 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐴) |
14 | nngt0 11656 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
15 | 14 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
16 | 9, 11, 13, 15 | divgt0d 11564 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < (𝐴 / 𝐵)) |
17 | 16 | biantrud 535 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)))) |
18 | elnnz 11979 | . . 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 314 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 < clt 10664 / cdiv 11286 ℕcn 11625 ℤcz 11969 ∥ cdvds 15599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-z 11970 df-dvds 15600 |
This theorem is referenced by: nndivides 15609 dvdsdivcl 15658 divgcdnn 15853 lcmgcdlem 15940 isprm6 16048 divnumden 16078 hashgcdlem 16115 hashgcdeq 16116 oddprmdvds 16229 gexexlem 18965 ablfac1lem 19183 pgpfac1lem3a 19191 fincygsubgodexd 19228 znrrg 20257 dvdsflf1o 25772 mersenne 25811 perfectlem1 25813 perfect 25815 dchrvmasumlem1 26079 dchrisum0flblem2 26093 logsqvma 26126 oddpwdc 31722 nndivdvdsd 39288 lcmineqlem4 39320 lcmineqlem23 39339 dffltz 39615 jm2.20nn 39938 jm2.27c 39948 fouriersw 42873 proththdlem 44131 perfectALTVlem1 44239 perfectALTV 44241 |
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