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Mirrors > Home > ILE Home > Th. List > dvdsflip | GIF version |
Description: An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.) |
Ref | Expression |
---|---|
dvdsflip.a | ⊢ 𝐴 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} |
dvdsflip.f | ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝑁 / 𝑦)) |
Ref | Expression |
---|---|
dvdsflip | ⊢ (𝑁 ∈ ℕ → 𝐹:𝐴–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsflip.f | . 2 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝑁 / 𝑦)) | |
2 | dvdsflip.a | . . . . 5 ⊢ 𝐴 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} | |
3 | 2 | eleq2i 2231 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
4 | dvdsdivcl 11773 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
5 | 3, 4 | sylan2b 285 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
6 | 5, 2 | eleqtrrdi 2258 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ 𝐴) |
7 | 2 | eleq2i 2231 | . . . 4 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
8 | dvdsdivcl 11773 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
9 | 7, 8 | sylan2b 285 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
10 | 9, 2 | eleqtrrdi 2258 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ 𝐴) |
11 | ssrab2 3222 | . . . . . . 7 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ ℕ | |
12 | 2, 11 | eqsstri 3169 | . . . . . 6 ⊢ 𝐴 ⊆ ℕ |
13 | 12 | sseli 3133 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℕ) |
14 | 12 | sseli 3133 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℕ) |
15 | 13, 14 | anim12i 336 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) |
16 | nncn 8856 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
17 | 16 | adantr 274 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑁 ∈ ℂ) |
18 | nncn 8856 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
19 | 18 | ad2antrl 482 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ∈ ℂ) |
20 | nncn 8856 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
21 | 20 | ad2antll 483 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ∈ ℂ) |
22 | simprr 522 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ∈ ℕ) | |
23 | 22 | nnap0d 8894 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 # 0) |
24 | 17, 19, 21, 23 | divmulap3d 8712 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ 𝑁 = (𝑦 · 𝑧))) |
25 | simprl 521 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ∈ ℕ) | |
26 | 25 | nnap0d 8894 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 # 0) |
27 | 17, 21, 19, 26 | divmulap2d 8711 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑦) = 𝑧 ↔ 𝑁 = (𝑦 · 𝑧))) |
28 | 24, 27 | bitr4d 190 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
29 | 15, 28 | sylan2 284 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
30 | eqcom 2166 | . . 3 ⊢ (𝑦 = (𝑁 / 𝑧) ↔ (𝑁 / 𝑧) = 𝑦) | |
31 | eqcom 2166 | . . 3 ⊢ (𝑧 = (𝑁 / 𝑦) ↔ (𝑁 / 𝑦) = 𝑧) | |
32 | 29, 30, 31 | 3bitr4g 222 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦 = (𝑁 / 𝑧) ↔ 𝑧 = (𝑁 / 𝑦))) |
33 | 1, 6, 10, 32 | f1o2d 6037 | 1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐴–1-1-onto→𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 {crab 2446 class class class wbr 3976 ↦ cmpt 4037 –1-1-onto→wf1o 5181 (class class class)co 5836 ℂcc 7742 · cmul 7749 / cdiv 8559 ℕcn 8848 ∥ cdvds 11713 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-dvds 11714 |
This theorem is referenced by: phisum 12149 |
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