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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 9100 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5767 · cmul 7618 ℤcz 9047 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: qapne 9424 qtri3or 10013 2tnp1ge0ge0 10067 flhalf 10068 intfracq 10086 zmodcl 10110 modqmul1 10143 addmodlteq 10164 sqoddm1div8 10437 eirraplem 11472 dvdscmulr 11511 dvdsmulcr 11512 modmulconst 11514 dvds2ln 11515 dvdsmod 11549 even2n 11560 2tp1odd 11570 ltoddhalfle 11579 m1expo 11586 m1exp1 11587 divalglemqt 11605 modremain 11615 flodddiv4 11620 gcdaddm 11661 gcdmultipled 11670 bezoutlemnewy 11673 bezoutlemstep 11674 bezoutlembi 11682 mulgcd 11693 dvdsmulgcd 11702 bezoutr 11709 lcmval 11733 lcmcllem 11737 lcmgcdlem 11747 mulgcddvds 11764 rpmulgcd2 11765 divgcdcoprm0 11771 cncongr1 11773 cncongr2 11774 prmind2 11790 exprmfct 11807 2sqpwodd 11843 hashdvds 11886 phimullem 11890 oddennn 11894 |
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