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Mirrors > Home > ILE Home > Th. List > zmulcld | GIF version |
Description: Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
zaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
Ref | Expression |
---|---|
zmulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | zaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
3 | zmulcl 9235 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 (class class class)co 5836 · cmul 7749 ℤcz 9182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 |
This theorem is referenced by: qapne 9568 qtri3or 10168 2tnp1ge0ge0 10226 flhalf 10227 intfracq 10245 zmodcl 10269 modqmul1 10302 addmodlteq 10323 sqoddm1div8 10597 eirraplem 11703 dvdscmulr 11746 dvdsmulcr 11747 modmulconst 11749 dvds2ln 11750 dvdsmod 11785 even2n 11796 2tp1odd 11806 ltoddhalfle 11815 m1expo 11822 m1exp1 11823 divalglemqt 11841 modremain 11851 flodddiv4 11856 gcdaddm 11902 gcdmultipled 11911 bezoutlemnewy 11914 bezoutlemstep 11915 bezoutlembi 11923 mulgcd 11934 dvdsmulgcd 11943 bezoutr 11950 lcmval 11974 lcmcllem 11978 lcmgcdlem 11988 mulgcddvds 12005 rpmulgcd2 12006 divgcdcoprm0 12012 cncongr1 12014 cncongr2 12015 prmind2 12031 exprmfct 12049 2sqpwodd 12087 hashdvds 12132 phimullem 12136 eulerthlem1 12138 eulerthlema 12141 eulerthlemh 12142 eulerthlemth 12143 prmdiv 12146 prmdiveq 12147 pythagtriplem2 12177 pythagtrip 12194 pcpremul 12204 pcqmul 12214 pcaddlem 12249 prmpwdvds 12264 oddennn 12268 |
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