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Mirrors > Home > ILE Home > Th. List > dvdsdivcl | GIF version |
Description: The complement of a divisor of 𝑁 is also a divisor of 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.) |
Ref | Expression |
---|---|
dvdsdivcl | ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3992 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∥ 𝑁 ↔ 𝐴 ∥ 𝑁)) | |
2 | 1 | elrab 2886 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ (𝐴 ∈ ℕ ∧ 𝐴 ∥ 𝑁)) |
3 | nndivdvds 11758 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐴 ∥ 𝑁 ↔ (𝑁 / 𝐴) ∈ ℕ)) | |
4 | 3 | biimpd 143 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐴 ∥ 𝑁 → (𝑁 / 𝐴) ∈ ℕ)) |
5 | 4 | expcom 115 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝑁 ∈ ℕ → (𝐴 ∥ 𝑁 → (𝑁 / 𝐴) ∈ ℕ))) |
6 | 5 | com23 78 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 ∥ 𝑁 → (𝑁 ∈ ℕ → (𝑁 / 𝐴) ∈ ℕ))) |
7 | 6 | imp 123 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∥ 𝑁) → (𝑁 ∈ ℕ → (𝑁 / 𝐴) ∈ ℕ)) |
8 | nnne0 8906 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) | |
9 | 8 | anim1i 338 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∥ 𝑁) → (𝐴 ≠ 0 ∧ 𝐴 ∥ 𝑁)) |
10 | 9 | ancomd 265 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∥ 𝑁) → (𝐴 ∥ 𝑁 ∧ 𝐴 ≠ 0)) |
11 | divconjdvds 11809 | . . . . . 6 ⊢ ((𝐴 ∥ 𝑁 ∧ 𝐴 ≠ 0) → (𝑁 / 𝐴) ∥ 𝑁) | |
12 | 10, 11 | syl 14 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∥ 𝑁) → (𝑁 / 𝐴) ∥ 𝑁) |
13 | 7, 12 | jctird 315 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∥ 𝑁) → (𝑁 ∈ ℕ → ((𝑁 / 𝐴) ∈ ℕ ∧ (𝑁 / 𝐴) ∥ 𝑁))) |
14 | 2, 13 | sylbi 120 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → (𝑁 ∈ ℕ → ((𝑁 / 𝐴) ∈ ℕ ∧ (𝑁 / 𝐴) ∥ 𝑁))) |
15 | 14 | impcom 124 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ((𝑁 / 𝐴) ∈ ℕ ∧ (𝑁 / 𝐴) ∥ 𝑁)) |
16 | breq1 3992 | . . 3 ⊢ (𝑥 = (𝑁 / 𝐴) → (𝑥 ∥ 𝑁 ↔ (𝑁 / 𝐴) ∥ 𝑁)) | |
17 | 16 | elrab 2886 | . 2 ⊢ ((𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↔ ((𝑁 / 𝐴) ∈ ℕ ∧ (𝑁 / 𝐴) ∥ 𝑁)) |
18 | 15, 17 | sylibr 133 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ≠ wne 2340 {crab 2452 class class class wbr 3989 (class class class)co 5853 0cc0 7774 / cdiv 8589 ℕcn 8878 ∥ cdvds 11749 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-n0 9136 df-z 9213 df-dvds 11750 |
This theorem is referenced by: dvdsflip 11811 |
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