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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 2833 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 4 | dvdsdivcl 16353 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
| 5 | 3, 4 | sylan2b 594 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 6 | 5, 2 | eleqtrrdi 2852 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐴) → (𝑁 / 𝑦) ∈ 𝐴) |
| 7 | 2 | eleq2i 2833 | . . . 4 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 8 | dvdsdivcl 16353 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | |
| 9 | 7, 8 | sylan2b 594 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 10 | 9, 2 | eleqtrrdi 2852 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝑁 / 𝑧) ∈ 𝐴) |
| 11 | 2 | ssrab3 4082 | . . . . . 6 ⊢ 𝐴 ⊆ ℕ |
| 12 | 11 | sseli 3979 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℕ) |
| 13 | 11 | sseli 3979 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℕ) |
| 14 | 12, 13 | anim12i 613 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) |
| 15 | nncn 12274 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
| 16 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑁 ∈ ℂ) |
| 17 | nncn 12274 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
| 18 | 17 | ad2antrl 728 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ∈ ℂ) |
| 19 | nncn 12274 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
| 20 | 19 | ad2antll 729 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ∈ ℂ) |
| 21 | nnne0 12300 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
| 22 | 21 | ad2antll 729 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑧 ≠ 0) |
| 23 | 16, 18, 20, 22 | divmul3d 12077 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ 𝑁 = (𝑦 · 𝑧))) |
| 24 | nnne0 12300 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
| 25 | 24 | ad2antrl 728 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → 𝑦 ≠ 0) |
| 26 | 16, 20, 18, 25 | divmul2d 12076 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑦) = 𝑧 ↔ 𝑁 = (𝑦 · 𝑧))) |
| 27 | 23, 26 | bitr4d 282 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑁 / 𝑧) = 𝑦 ↔ (𝑁 / 𝑦) = 𝑧)) |
| 28 | 14, 27 | sylan2 593 | . . 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 314 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦 = (𝑁 / 𝑧) ↔ 𝑧 = (𝑁 / 𝑦))) |
| 32 | 1, 6, 10, 31 | f1o2d 7687 | 1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐴–1-1-onto→𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 class class class wbr 5143 ↦ cmpt 5225 –1-1-onto→wf1o 6560 (class class class)co 7431 ℂcc 11153 0cc0 11155 · cmul 11160 / cdiv 11920 ℕcn 12266 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-z 12614 df-dvds 16291 |
| This theorem is referenced by: phisum 16828 fsumdvdscom 27228 |
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