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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 11259 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mulid2d 9937 | . 2 ⊢ (𝑁 ∈ ℤ → (1 · 𝑁) = 𝑁) |
3 | 1z 11284 | . . . 4 ⊢ 1 ∈ ℤ | |
4 | dvds0lem 14830 | . . . 4 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) | |
5 | 3, 4 | mp3anl1 1410 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) |
6 | 5 | anabsan 850 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) |
7 | 2, 6 | mpdan 699 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 1c1 9816 · cmul 9820 ℤcz 11254 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-z 11255 df-dvds 14822 |
This theorem is referenced by: dvdsadd 14862 dvds1 14879 dvdsext 14881 z2even 14944 n2dvds3 14945 divalglem0 14954 divalglem2 14956 sadadd3 15021 gcd0id 15078 gcdzeq 15109 mulgcddvds 15207 1idssfct 15231 dvdsprime 15238 3prm 15244 dvdsprm 15253 exprmfct 15254 coprm 15261 isprm6 15264 pcidlem 15414 pcprmpw2 15424 pcprmpw 15425 prmgaplem1 15591 prmgaplem2 15592 prmgaplcmlem1 15593 prmgaplcmlem2 15594 odeq 17792 pgpfi 17843 znidomb 19729 sgmnncl 24673 muinv 24719 ppiublem2 24728 perfect1 24753 perfectlem2 24755 2lgslem2 24920 2lgs2 24930 2sqlem6 24948 eupath2lem3 26506 ex-ind-dvds 26710 eulerpartlemt 29760 poimirlem25 32604 poimirlem27 32606 jm2.18 36573 jm2.15nn0 36588 jm2.16nn0 36589 jm2.27c 36592 nzss 37538 etransclem25 39152 perfectALTVlem2 40165 eupth2lem3lem3 41398 |
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