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Mirrors > Home > ILE Home > Th. List > flqpmodeq | GIF version |
Description: Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Ref | Expression |
---|---|
flqpmodeq | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqvalr 9881 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵))) | |
2 | 1 | eqcomd 2100 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)) = (𝐴 mod 𝐵)) |
3 | qcn 9218 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
4 | 3 | 3ad2ant1 967 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐴 ∈ ℂ) |
5 | simp3 948 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 < 𝐵) | |
6 | 5 | gt0ne0d 8087 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ≠ 0) |
7 | qdivcl 9227 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) | |
8 | 6, 7 | syld3an3 1226 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℚ) |
9 | 8 | flqcld 9833 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
10 | 9 | zcnd 8968 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℂ) |
11 | qcn 9218 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
12 | 11 | 3ad2ant2 968 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℂ) |
13 | 10, 12 | mulcld 7605 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((⌊‘(𝐴 / 𝐵)) · 𝐵) ∈ ℂ) |
14 | modqcl 9882 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ) | |
15 | qcn 9218 | . . . 4 ⊢ ((𝐴 mod 𝐵) ∈ ℚ → (𝐴 mod 𝐵) ∈ ℂ) | |
16 | 14, 15 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℂ) |
17 | 4, 13, 16 | subaddd 7908 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)) = (𝐴 mod 𝐵) ↔ (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴)) |
18 | 2, 17 | mpbid 146 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ≠ wne 2262 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 ℂcc 7445 0cc0 7447 + caddc 7450 · cmul 7452 < clt 7619 − cmin 7750 / cdiv 8236 ℚcq 9203 ⌊cfl 9824 mod cmo 9878 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-n0 8772 df-z 8849 df-q 9204 df-rp 9234 df-fl 9826 df-mod 9879 |
This theorem is referenced by: modqmuladd 9922 |
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