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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 8699 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 403 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 (class class class)co 5591 · cmul 7258 ℤcz 8646 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 |
This theorem is referenced by: qapne 9019 qtri3or 9543 2tnp1ge0ge0 9597 flhalf 9598 intfracq 9616 zmodcl 9640 modqmul1 9673 addmodlteq 9694 sqoddm1div8 9941 dvdscmulr 10605 dvdsmulcr 10606 modmulconst 10608 dvds2ln 10609 dvdsmod 10643 even2n 10654 2tp1odd 10664 ltoddhalfle 10673 m1expo 10680 m1exp1 10681 divalglemqt 10699 modremain 10709 flodddiv4 10714 gcdaddm 10755 bezoutlemnewy 10765 bezoutlemstep 10766 bezoutlembi 10774 mulgcd 10785 dvdsmulgcd 10794 bezoutr 10801 lcmval 10825 lcmcllem 10829 lcmgcdlem 10839 mulgcddvds 10856 rpmulgcd2 10857 divgcdcoprm0 10863 cncongr1 10865 cncongr2 10866 prmind2 10882 exprmfct 10899 2sqpwodd 10934 hashdvds 10977 phimullem 10981 oddennn 10985 |
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