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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 12615 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mulridd 11275 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 · 1) = 𝑁) |
3 | 1z 12644 | . . . 4 ⊢ 1 ∈ ℤ | |
4 | dvds0lem 16300 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 · 1) = 𝑁) → 1 ∥ 𝑁) | |
5 | 3, 4 | mp3anl2 1455 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 · 1) = 𝑁) → 1 ∥ 𝑁) |
6 | 5 | anabsan 665 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 · 1) = 𝑁) → 1 ∥ 𝑁) |
7 | 2, 6 | mpdan 687 | 1 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 1c1 11153 · cmul 11157 ℤcz 12610 ∥ cdvds 16286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rrecex 11224 ax-cnre 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-neg 11492 df-nn 12264 df-z 12611 df-dvds 16287 |
This theorem is referenced by: dvds1 16352 gcdcllem1 16532 gcdcllem3 16534 lcmfunsnlem 16674 coprmproddvds 16696 1idssfct 16713 isprm2lem 16714 dvdsprime 16720 pclem 16871 prmreclem1 16949 oddvdssubg 19887 perfectlem2 27288 oddpwdc 34335 perfectALTVlem2 47646 |
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