Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2901 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
4 | dvdsdivcl 15654 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
5 | 3, 4 | sylan2b 593 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
6 | 5, 2 | eleqtrrdi 2921 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ 𝐴) |
7 | 2 | eleq2i 2901 | . . . 4 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
8 | dvdsdivcl 15654 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
9 | 7, 8 | sylan2b 593 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
10 | 9, 2 | eleqtrrdi 2921 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ 𝐴) |
11 | 2 | ssrab3 4054 | . . . . . 6 ⊢ 𝐴 ⊆ ℕ |
12 | 11 | sseli 3960 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℕ) |
13 | 11 | sseli 3960 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℕ) |
14 | 12, 13 | anim12i 612 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) |
15 | nncn 11634 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
16 | 15 | adantr 481 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑁 ∈ ℂ) |
17 | nncn 11634 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | 17 | ad2antrl 724 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ∈ ℂ) |
19 | nncn 11634 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
20 | 19 | ad2antll 725 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ∈ ℂ) |
21 | nnne0 11659 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
22 | 21 | ad2antll 725 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ≠ 0) |
23 | 16, 18, 20, 22 | divmul3d 11438 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ 𝑁 = (𝑦 · 𝑧))) |
24 | nnne0 11659 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
25 | 24 | ad2antrl 724 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ≠ 0) |
26 | 16, 20, 18, 25 | divmul2d 11437 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑦) = 𝑧 ↔ 𝑁 = (𝑦 · 𝑧))) |
27 | 23, 26 | bitr4d 283 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
28 | 14, 27 | sylan2 592 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
29 | eqcom 2825 | . . 3 ⊢ (𝑦 = (𝑁 / 𝑧) ↔ (𝑁 / 𝑧) = 𝑦) | |
30 | eqcom 2825 | . . 3 ⊢ (𝑧 = (𝑁 / 𝑦) ↔ (𝑁 / 𝑦) = 𝑧) | |
31 | 28, 29, 30 | 3bitr4g 315 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦 = (𝑁 / 𝑧) ↔ 𝑧 = (𝑁 / 𝑦))) |
32 | 1, 6, 10, 31 | f1o2d 7388 | 1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐴–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 class class class wbr 5057 ↦ cmpt 5137 –1-1-onto→wf1o 6347 (class class class)co 7145 ℂcc 10523 0cc0 10525 · cmul 10530 / cdiv 11285 ℕcn 11626 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-z 11970 df-dvds 15596 |
This theorem is referenced by: phisum 16115 fsumdvdscom 25689 |
Copyright terms: Public domain | W3C validator |