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Mirrors > Home > MPE Home > Th. List > dvdsssfz1 | Structured version Visualization version GIF version |
Description: The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
dvdsssfz1 | ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 12005 | . . . . 5 ⊢ (𝑝 ∈ ℕ → 𝑝 ∈ ℤ) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ) | |
3 | dvdsle 15660 | . . . . 5 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (𝑝 ∥ 𝐴 → 𝑝 ≤ 𝐴)) | |
4 | 1, 2, 3 | syl2anr 598 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑝 ∥ 𝐴 → 𝑝 ≤ 𝐴)) |
5 | ibar 531 | . . . . . 6 ⊢ (𝑝 ∈ ℕ → (𝑝 ≤ 𝐴 ↔ (𝑝 ∈ ℕ ∧ 𝑝 ≤ 𝐴))) | |
6 | 5 | adantl 484 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑝 ≤ 𝐴 ↔ (𝑝 ∈ ℕ ∧ 𝑝 ≤ 𝐴))) |
7 | nnz 12005 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
8 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℕ) → 𝐴 ∈ ℤ) |
9 | fznn 12976 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ (1...𝐴) ↔ (𝑝 ∈ ℕ ∧ 𝑝 ≤ 𝐴))) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑝 ∈ (1...𝐴) ↔ (𝑝 ∈ ℕ ∧ 𝑝 ≤ 𝐴))) |
11 | 6, 10 | bitr4d 284 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑝 ≤ 𝐴 ↔ 𝑝 ∈ (1...𝐴))) |
12 | 4, 11 | sylibd 241 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑝 ∥ 𝐴 → 𝑝 ∈ (1...𝐴))) |
13 | 12 | ralrimiva 3182 | . 2 ⊢ (𝐴 ∈ ℕ → ∀𝑝 ∈ ℕ (𝑝 ∥ 𝐴 → 𝑝 ∈ (1...𝐴))) |
14 | rabss 4048 | . 2 ⊢ ({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴) ↔ ∀𝑝 ∈ ℕ (𝑝 ∥ 𝐴 → 𝑝 ∈ (1...𝐴))) | |
15 | 13, 14 | sylibr 236 | 1 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 {crab 3142 ⊆ wss 3936 class class class wbr 5066 (class class class)co 7156 1c1 10538 ≤ cle 10676 ℕcn 11638 ℤcz 11982 ...cfz 12893 ∥ cdvds 15607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-dvds 15608 |
This theorem is referenced by: phisum 16127 prmdvdsfi 25684 0sgm 25721 sgmf 25722 sgmnncl 25724 mumul 25758 sqff1o 25759 fsumdvdsdiag 25761 fsumdvdscom 25762 dvdsflsumcom 25765 musum 25768 musumsum 25769 muinv 25770 fsumdvdsmul 25772 vmasum 25792 perfectlem2 25806 dchrvmasumlem1 26071 dchrisum0ff 26083 dchrisum0 26096 vmalogdivsum2 26114 logsqvma 26118 logsqvma2 26119 selberg 26124 selberg34r 26147 pntsval2 26152 pntrlog2bndlem1 26153 perfectALTVlem2 43907 |
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