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Mirrors > Home > MPE Home > Th. List > nndivdvds | Structured version Visualization version GIF version |
Description: Strong form of dvdsval2 16207 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 12583 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
2 | nnne0 12250 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
3 | nnz 12583 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ) |
5 | dvdsval2 16207 | . . . 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 12223 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | nnre 12223 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
11 | 10 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
12 | nngt0 12247 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐴) |
14 | nngt0 12247 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
15 | 14 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
16 | 9, 11, 13, 15 | divgt0d 12153 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < (𝐴 / 𝐵)) |
17 | 16 | biantrud 531 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)))) |
18 | elnnz 12572 | . . 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 311 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 (class class class)co 7405 ℝcr 11111 0cc0 11112 < clt 11252 / cdiv 11875 ℕcn 12216 ℤcz 12562 ∥ cdvds 16204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-z 12563 df-dvds 16205 |
This theorem is referenced by: nndivides 16214 dvdsdivcl 16266 divgcdnn 16463 lcmgcdlem 16550 isprm6 16658 divnumden 16693 hashgcdlem 16730 hashgcdeq 16731 oddprmdvds 16845 gexexlem 19772 ablfac1lem 19990 pgpfac1lem3a 19998 fincygsubgodexd 20035 znrrg 21460 dvdsflf1o 27074 mersenne 27115 perfectlem1 27117 perfect 27119 dchrvmasumlem1 27383 dchrisum0flblem2 27397 logsqvma 27430 oddpwdc 33883 nndivdvdsd 41381 lcmineqlem4 41413 lcmineqlem23 41432 aks6d1c1p3 41487 aks6d1c2p1 41495 aks6d1c2p2 41496 dffltz 41954 jm2.20nn 42314 jm2.27c 42324 fouriersw 45519 proththdlem 46853 perfectALTVlem1 46961 perfectALTV 46963 |
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