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| Mirrors > Home > MPE Home > Th. List > iddvds | Structured version Visualization version GIF version | ||
| Description: An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
| Ref | Expression |
|---|---|
| iddvds | ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zcn 12618 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
| 2 | 1 | mullidd 11279 | . 2 ⊢ (𝑁 ∈ ℤ → (1 · 𝑁) = 𝑁) |
| 3 | 1z 12647 | . . . 4 ⊢ 1 ∈ ℤ | |
| 4 | dvds0lem 16304 | . . . 4 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) | |
| 5 | 3, 4 | mp3anl1 1457 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) |
| 6 | 5 | anabsan 665 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) |
| 7 | 2, 6 | mpdan 687 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 1c1 11156 · cmul 11160 ℤcz 12613 ∥ 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-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-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rrecex 11227 ax-cnre 11228 |
| 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-ral 3062 df-rex 3071 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-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-neg 11495 df-nn 12267 df-z 12614 df-dvds 16291 |
| This theorem is referenced by: dvdsadd 16339 dvds1 16356 dvdsext 16358 z2even 16407 divalglem0 16430 divalglem2 16432 sadadd3 16498 gcd0id 16556 gcdzeq 16589 mulgcddvds 16692 1idssfct 16717 isprm2lem 16718 dvdsprime 16724 dvdsprm 16740 exprmfct 16741 coprm 16748 isprm6 16751 pcidlem 16910 pcprmpw2 16920 pcprmpw 16921 prmgaplem1 17087 prmgaplem2 17088 prmgaplcmlem1 17089 prmgaplcmlem2 17090 odeq 19568 pgpfi 19623 znidomb 21580 sgmnncl 27190 muinv 27236 ppiublem2 27247 perfect1 27272 perfectlem2 27274 2sqlem6 27467 ex-ind-dvds 30480 eulerpartlemt 34373 dfgcd3 37325 poimirlem25 37652 poimirlem27 37654 aks4d1p9 42089 unitscyglem2 42197 unitscyglem4 42199 jm2.18 43000 jm2.15nn0 43015 jm2.16nn0 43016 jm2.27c 43019 nzss 44336 etransclem25 46274 perfectALTVlem2 47709 |
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