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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 9620 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ) | |
5 | 1, 2, 3, 4 | syl3anc 1170 | 1 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5589 0cc0 7251 < clt 7423 ℚcq 8997 mod cmo 9616 |
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-13 1445 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 3999 ax-un 4223 ax-setind 4315 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 ax-arch 7365 |
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-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 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-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-po 4086 df-iso 4087 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-n0 8564 df-z 8645 df-q 8998 df-rp 9028 df-fl 9564 df-mod 9617 |
This theorem is referenced by: modqabs 9651 modqaddabs 9656 modqaddmod 9657 modqmuladdim 9661 modqmuladdnn0 9662 modqadd2mod 9668 modqltm1p1mod 9670 modqsubmod 9676 modqsubmodmod 9677 modqmulmod 9683 modqmulmodr 9684 modqaddmulmod 9685 modqsubdir 9687 addmodlteq 9692 divalgmod 10705 bezoutlemnewy 10763 crth 10978 phimullem 10979 |
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