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Mirrors > Home > ILE Home > Th. List > nndiv | GIF version |
Description: Two ways to express "𝐴 divides 𝐵 " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nndiv | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 2407 | . 2 ⊢ ((𝐵 / 𝐴) ∈ ℕ ↔ ∃𝑥 ∈ ℕ 𝑥 = (𝐵 / 𝐴)) | |
2 | eqcom 2091 | . . . 4 ⊢ (𝑥 = (𝐵 / 𝐴) ↔ (𝐵 / 𝐴) = 𝑥) | |
3 | nncn 8491 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ) | |
4 | 3 | ad2antlr 474 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝐵 ∈ ℂ) |
5 | nncn 8491 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
6 | 5 | ad2antrr 473 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ) |
7 | nncn 8491 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
8 | 7 | adantl 272 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ) |
9 | nnap0 8512 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) | |
10 | 9 | ad2antrr 473 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝐴 # 0) |
11 | 4, 6, 8, 10 | divmulapd 8340 | . . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((𝐵 / 𝐴) = 𝑥 ↔ (𝐴 · 𝑥) = 𝐵)) |
12 | 2, 11 | syl5bb 191 | . . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → (𝑥 = (𝐵 / 𝐴) ↔ (𝐴 · 𝑥) = 𝐵)) |
13 | 12 | rexbidva 2378 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ 𝑥 = (𝐵 / 𝐴) ↔ ∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵)) |
14 | 1, 13 | syl5rbb 192 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 ∃wrex 2361 class class class wbr 3851 (class class class)co 5666 ℂcc 7409 0cc0 7411 · cmul 7416 # cap 8119 / cdiv 8200 ℕcn 8483 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 df-inn 8484 |
This theorem is referenced by: nndivides 11142 |
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