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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 8404 | . 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 1433 (class class class)co 5532 · cmul 6986 ℤcz 8351 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: qapne 8724 qtri3or 9252 2tnp1ge0ge0 9303 flhalf 9304 intfracq 9322 zmodcl 9346 modqmul1 9379 addmodlteq 9400 sqoddm1div8 9625 dvdscmulr 10224 dvdsmulcr 10225 modmulconst 10227 dvds2ln 10228 dvdsmod 10262 even2n 10273 2tp1odd 10284 ltoddhalfle 10293 m1expo 10300 m1exp1 10301 divalglemqt 10319 modremain 10329 flodddiv4 10334 gcdaddm 10375 bezoutlemnewy 10385 bezoutlemstep 10386 bezoutlembi 10394 mulgcd 10405 dvdsmulgcd 10414 bezoutr 10421 lcmval 10445 lcmcllem 10449 lcmgcdlem 10459 mulgcddvds 10476 rpmulgcd2 10477 divgcdcoprm0 10483 cncongr1 10485 cncongr2 10486 prmind2 10502 exprmfct 10519 2sqpwodd 10554 |
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