Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 0dvds | GIF version |
Description: Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
0dvds | ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 9033 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | divides 11422 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) | |
3 | 1, 2 | mpan 420 | . . 3 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) |
4 | zcn 9027 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
5 | 4 | mul01d 8123 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 0) = 0) |
6 | eqtr2 2136 | . . . . . 6 ⊢ (((𝑛 · 0) = 𝑁 ∧ (𝑛 · 0) = 0) → 𝑁 = 0) | |
7 | 5, 6 | sylan2 284 | . . . . 5 ⊢ (((𝑛 · 0) = 𝑁 ∧ 𝑛 ∈ ℤ) → 𝑁 = 0) |
8 | 7 | ancoms 266 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 · 0) = 𝑁) → 𝑁 = 0) |
9 | 8 | rexlimiva 2521 | . . 3 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁 → 𝑁 = 0) |
10 | 3, 9 | syl6bi 162 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 → 𝑁 = 0)) |
11 | dvds0 11435 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
12 | 1, 11 | ax-mp 5 | . . 3 ⊢ 0 ∥ 0 |
13 | breq2 3903 | . . 3 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0)) | |
14 | 12, 13 | mpbiri 167 | . 2 ⊢ (𝑁 = 0 → 0 ∥ 𝑁) |
15 | 10, 14 | impbid1 141 | 1 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 class class class wbr 3899 (class class class)co 5742 0cc0 7588 · cmul 7593 ℤcz 9022 ∥ cdvds 11420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 df-z 9023 df-dvds 11421 |
This theorem is referenced by: zdvdsdc 11441 dvdsabseq 11472 bezoutlemle 11623 dfgcd3 11625 dfgcd2 11629 dvdssq 11646 rpdvds 11707 |
Copyright terms: Public domain | W3C validator |