Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dvdssubr | GIF version |
Description: An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Ref | Expression |
---|---|
dvdssubr | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ 𝑀 ∥ (𝑁 − 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsubcl 8392 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ) | |
2 | 1 | ancoms 264 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ) |
3 | dvdsadd 10238 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 𝑀) ∈ ℤ) → (𝑀 ∥ (𝑁 − 𝑀) ↔ 𝑀 ∥ (𝑀 + (𝑁 − 𝑀)))) | |
4 | 2, 3 | syldan 276 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑁 − 𝑀) ↔ 𝑀 ∥ (𝑀 + (𝑁 − 𝑀)))) |
5 | zcn 8356 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
6 | zcn 8356 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
7 | pncan3 7316 | . . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 + (𝑁 − 𝑀)) = 𝑁) | |
8 | 5, 6, 7 | syl2an 283 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + (𝑁 − 𝑀)) = 𝑁) |
9 | 8 | breq2d 3797 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 + (𝑁 − 𝑀)) ↔ 𝑀 ∥ 𝑁)) |
10 | 4, 9 | bitr2d 187 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ 𝑀 ∥ (𝑁 − 𝑀))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℂcc 6979 + caddc 6984 − cmin 7279 ℤcz 8351 ∥ cdvds 10195 |
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-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-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 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-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-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-inn 8040 df-n0 8289 df-z 8352 df-dvds 10196 |
This theorem is referenced by: ndvdsadd 10331 |
Copyright terms: Public domain | W3C validator |