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Mirrors > Home > ILE Home > Th. List > nndivdvds | GIF version |
Description: Strong form of dvdsval2 10579 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 8665 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
2 | 1 | adantl 271 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ) |
3 | nnne0 8344 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
4 | 3 | adantl 271 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0) |
5 | nnz 8665 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
6 | 5 | adantr 270 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ) |
7 | dvdsval2 10579 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℤ)) | |
8 | 2, 4, 6, 7 | syl3anc 1170 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℤ)) |
9 | 8 | anbi1d 453 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)) ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
10 | nnre 8323 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
11 | 10 | adantr 270 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ) |
12 | nnre 8323 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
13 | 12 | adantl 271 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
14 | nngt0 8341 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
15 | 14 | adantr 270 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐴) |
16 | nngt0 8341 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
17 | 16 | adantl 271 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
18 | 11, 13, 15, 17 | divgt0d 8290 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 0 < (𝐴 / 𝐵)) |
19 | 18 | biantrud 298 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐵 ∥ 𝐴 ∧ 0 < (𝐴 / 𝐵)))) |
20 | elnnz 8656 | . . 3 ⊢ ((𝐴 / 𝐵) ∈ ℕ ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵))) | |
21 | 20 | a1i 9 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ ↔ ((𝐴 / 𝐵) ∈ ℤ ∧ 0 < (𝐴 / 𝐵)))) |
22 | 9, 19, 21 | 3bitr4d 218 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1434 ≠ wne 2249 class class class wbr 3811 (class class class)co 5591 ℝcr 7252 0cc0 7253 < clt 7425 / cdiv 8037 ℕcn 8316 ℤcz 8646 ∥ cdvds 10576 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-po 4087 df-iso 4088 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-n0 8566 df-z 8647 df-dvds 10577 |
This theorem is referenced by: nndivides 10583 dvdsdivcl 10631 divgcdnn 10746 lcmgcdlem 10839 isprm6 10906 oddpwdclemodd 10930 oddpwdclemdc 10931 divnumden 10954 hashgcdlem 10983 hashgcdeq 10984 |
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