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Mirrors > Home > MPE Home > Th. List > dvdsflip | Structured version Visualization version 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 2829 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
4 | dvdsdivcl 15877 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
5 | 3, 4 | sylan2b 597 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
6 | 5, 2 | eleqtrrdi 2849 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ 𝐴) |
7 | 2 | eleq2i 2829 | . . . 4 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
8 | dvdsdivcl 15877 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
9 | 7, 8 | sylan2b 597 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
10 | 9, 2 | eleqtrrdi 2849 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ 𝐴) |
11 | 2 | ssrab3 3995 | . . . . . 6 ⊢ 𝐴 ⊆ ℕ |
12 | 11 | sseli 3896 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℕ) |
13 | 11 | sseli 3896 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℕ) |
14 | 12, 13 | anim12i 616 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) |
15 | nncn 11838 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
16 | 15 | adantr 484 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑁 ∈ ℂ) |
17 | nncn 11838 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | 17 | ad2antrl 728 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ∈ ℂ) |
19 | nncn 11838 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
20 | 19 | ad2antll 729 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ∈ ℂ) |
21 | nnne0 11864 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
22 | 21 | ad2antll 729 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ≠ 0) |
23 | 16, 18, 20, 22 | divmul3d 11642 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ 𝑁 = (𝑦 · 𝑧))) |
24 | nnne0 11864 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
25 | 24 | ad2antrl 728 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ≠ 0) |
26 | 16, 20, 18, 25 | divmul2d 11641 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑦) = 𝑧 ↔ 𝑁 = (𝑦 · 𝑧))) |
27 | 23, 26 | bitr4d 285 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
28 | 14, 27 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
29 | eqcom 2744 | . . 3 ⊢ (𝑦 = (𝑁 / 𝑧) ↔ (𝑁 / 𝑧) = 𝑦) | |
30 | eqcom 2744 | . . 3 ⊢ (𝑧 = (𝑁 / 𝑦) ↔ (𝑁 / 𝑦) = 𝑧) | |
31 | 28, 29, 30 | 3bitr4g 317 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦 = (𝑁 / 𝑧) ↔ 𝑧 = (𝑁 / 𝑦))) |
32 | 1, 6, 10, 31 | f1o2d 7459 | 1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐴–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {crab 3065 class class class wbr 5053 ↦ cmpt 5135 –1-1-onto→wf1o 6379 (class class class)co 7213 ℂcc 10727 0cc0 10729 · cmul 10734 / cdiv 11489 ℕcn 11830 ∥ cdvds 15815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-z 12177 df-dvds 15816 |
This theorem is referenced by: phisum 16343 fsumdvdscom 26067 |
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