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Mirrors > Home > ILE Home > Th. List > modqcld | GIF version |
Description: Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Ref | Expression |
---|---|
modqcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℚ) |
modqcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℚ) |
modqcld.3 | ⊢ (𝜑 → 0 < 𝐵) |
Ref | Expression |
---|---|
modqcld | ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℚ) | |
2 | modqcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℚ) | |
3 | modqcld.3 | . 2 ⊢ (𝜑 → 0 < 𝐵) | |
4 | modqcl 10269 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ) | |
5 | 1, 2, 3, 4 | syl3anc 1233 | 1 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 0cc0 7761 < clt 7941 ℚcq 9565 mod cmo 10265 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-n0 9123 df-z 9200 df-q 9566 df-rp 9598 df-fl 10213 df-mod 10266 |
This theorem is referenced by: modqabs 10300 modqaddabs 10305 modqaddmod 10306 modqmuladdim 10310 modqmuladdnn0 10311 modqadd2mod 10317 modqltm1p1mod 10319 modqsubmod 10325 modqsubmodmod 10326 modqmulmod 10332 modqmulmodr 10333 modqaddmulmod 10334 modqsubdir 10336 addmodlteq 10341 divalgmod 11873 bezoutlemnewy 11938 crth 12165 phimullem 12166 4sqlem5 12321 4sqlem6 12322 4sqlem10 12326 |
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