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Mirrors > Home > ILE Home > Th. List > negqmod0 | GIF version |
Description: 𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
negqmod0 | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qcn 9012 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
2 | 1 | 3ad2ant1 960 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐴 ∈ ℂ) |
3 | qcn 9012 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
4 | 3 | 3ad2ant2 961 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℂ) |
5 | qre 9003 | . . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
6 | 5 | 3ad2ant2 961 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
7 | simp3 941 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 < 𝐵) | |
8 | 6, 7 | gt0ap0d 8003 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 # 0) |
9 | 2, 4, 8 | divclapd 8152 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℂ) |
10 | znegclb 8677 | . . . 4 ⊢ ((𝐴 / 𝐵) ∈ ℂ → ((𝐴 / 𝐵) ∈ ℤ ↔ -(𝐴 / 𝐵) ∈ ℤ)) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) ∈ ℤ ↔ -(𝐴 / 𝐵) ∈ ℤ)) |
12 | 2, 4, 8 | divnegapd 8165 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
13 | 12 | eleq1d 2151 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (-(𝐴 / 𝐵) ∈ ℤ ↔ (-𝐴 / 𝐵) ∈ ℤ)) |
14 | 11, 13 | bitrd 186 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) ∈ ℤ ↔ (-𝐴 / 𝐵) ∈ ℤ)) |
15 | modq0 9623 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ)) | |
16 | qnegcl 9014 | . . 3 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ) | |
17 | modq0 9623 | . . 3 ⊢ ((-𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((-𝐴 mod 𝐵) = 0 ↔ (-𝐴 / 𝐵) ∈ ℤ)) | |
18 | 16, 17 | syl3an1 1203 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((-𝐴 mod 𝐵) = 0 ↔ (-𝐴 / 𝐵) ∈ ℤ)) |
19 | 14, 15, 18 | 3bitr4d 218 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5589 ℂcc 7249 ℝcr 7250 0cc0 7251 < clt 7423 -cneg 7555 / cdiv 8035 ℤcz 8644 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: (None) |
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