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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 9011 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 406 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 (class class class)co 5728 · cmul 7552 ℤcz 8958 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-sub 7858 df-neg 7859 df-inn 8631 df-n0 8882 df-z 8959 |
This theorem is referenced by: qapne 9333 qtri3or 9913 2tnp1ge0ge0 9967 flhalf 9968 intfracq 9986 zmodcl 10010 modqmul1 10043 addmodlteq 10064 sqoddm1div8 10337 eirraplem 11331 dvdscmulr 11370 dvdsmulcr 11371 modmulconst 11373 dvds2ln 11374 dvdsmod 11408 even2n 11419 2tp1odd 11429 ltoddhalfle 11438 m1expo 11445 m1exp1 11446 divalglemqt 11464 modremain 11474 flodddiv4 11479 gcdaddm 11520 bezoutlemnewy 11530 bezoutlemstep 11531 bezoutlembi 11539 mulgcd 11550 dvdsmulgcd 11559 bezoutr 11566 lcmval 11590 lcmcllem 11594 lcmgcdlem 11604 mulgcddvds 11621 rpmulgcd2 11622 divgcdcoprm0 11628 cncongr1 11630 cncongr2 11631 prmind2 11647 exprmfct 11664 2sqpwodd 11699 hashdvds 11742 phimullem 11746 oddennn 11750 |
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