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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 2394 | . 2 ⊢ ((𝐵 / 𝐴) ∈ ℕ ↔ ∃𝑥 ∈ ℕ 𝑥 = (𝐵 / 𝐴)) | |
2 | eqcom 2083 | . . . 4 ⊢ (𝑥 = (𝐵 / 𝐴) ↔ (𝐵 / 𝐴) = 𝑥) | |
3 | nncn 8047 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ) | |
4 | 3 | ad2antlr 472 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝐵 ∈ ℂ) |
5 | nncn 8047 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
6 | 5 | ad2antrr 471 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ) |
7 | nncn 8047 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
8 | 7 | adantl 271 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ) |
9 | nnap0 8068 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) | |
10 | 9 | ad2antrr 471 | . . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → 𝐴 # 0) |
11 | 4, 6, 8, 10 | divmulapd 7899 | . . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((𝐵 / 𝐴) = 𝑥 ↔ (𝐴 · 𝑥) = 𝐵)) |
12 | 2, 11 | syl5bb 190 | . . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → (𝑥 = (𝐵 / 𝐴) ↔ (𝐴 · 𝑥) = 𝐵)) |
13 | 12 | rexbidva 2365 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ 𝑥 = (𝐵 / 𝐴) ↔ ∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵)) |
14 | 1, 13 | syl5rbb 191 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 class class class wbr 3785 (class class class)co 5532 ℂcc 6979 0cc0 6981 · cmul 6986 # cap 7681 / cdiv 7760 ℕcn 8039 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 |
This theorem is referenced by: nndivides 10202 |
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