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Mirrors > Home > MPE Home > Th. List > 1dvds | Structured version Visualization version GIF version |
Description: 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
1dvds | ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 11989 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mulid1d 10660 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 · 1) = 𝑁) |
3 | 1z 12015 | . . . 4 ⊢ 1 ∈ ℤ | |
4 | dvds0lem 15622 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 · 1) = 𝑁) → 1 ∥ 𝑁) | |
5 | 3, 4 | mp3anl2 1452 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 · 1) = 𝑁) → 1 ∥ 𝑁) |
6 | 5 | anabsan 663 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 · 1) = 𝑁) → 1 ∥ 𝑁) |
7 | 2, 6 | mpdan 685 | 1 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 1c1 10540 · cmul 10544 ℤcz 11984 ∥ cdvds 15609 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rrecex 10611 ax-cnre 10612 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-neg 10875 df-nn 11641 df-z 11985 df-dvds 15610 |
This theorem is referenced by: dvds1 15671 gcdcllem1 15850 gcdcllem3 15852 lcmfunsnlem 15987 coprmproddvds 16009 1idssfct 16026 isprm2lem 16027 dvdsprime 16033 pclem 16177 prmreclem1 16254 oddvdssubg 18977 perfectlem2 25808 oddpwdc 31614 perfectALTVlem2 43894 |
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