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Mirrors > Home > ILE Home > Th. List > modqnegd | GIF version |
Description: Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.) |
Ref | Expression |
---|---|
modqnegd.1 | ⊢ (𝜑 → 𝐴 ∈ ℚ) |
modqnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℚ) |
modqnegd.3 | ⊢ (𝜑 → 𝐶 ∈ ℚ) |
modqnegd.cgt0 | ⊢ (𝜑 → 0 < 𝐶) |
modqnegd.4 | ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶)) |
Ref | Expression |
---|---|
modqnegd | ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqnegd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ) | |
2 | modqnegd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℚ) | |
3 | neg1z 9086 | . . . 4 ⊢ -1 ∈ ℤ | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝜑 → -1 ∈ ℤ) |
5 | modqnegd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℚ) | |
6 | modqnegd.cgt0 | . . 3 ⊢ (𝜑 → 0 < 𝐶) | |
7 | modqnegd.4 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶)) | |
8 | 1, 2, 4, 5, 6, 7 | modqmul1 10150 | . 2 ⊢ (𝜑 → ((𝐴 · -1) mod 𝐶) = ((𝐵 · -1) mod 𝐶)) |
9 | qcn 9426 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
10 | 1, 9 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | 4 | zcnd 9174 | . . . . 5 ⊢ (𝜑 → -1 ∈ ℂ) |
12 | 10, 11 | mulcomd 7787 | . . . 4 ⊢ (𝜑 → (𝐴 · -1) = (-1 · 𝐴)) |
13 | 10 | mulm1d 8172 | . . . 4 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
14 | 12, 13 | eqtrd 2172 | . . 3 ⊢ (𝜑 → (𝐴 · -1) = -𝐴) |
15 | 14 | oveq1d 5789 | . 2 ⊢ (𝜑 → ((𝐴 · -1) mod 𝐶) = (-𝐴 mod 𝐶)) |
16 | qcn 9426 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
17 | 2, 16 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
18 | 17, 11 | mulcomd 7787 | . . . 4 ⊢ (𝜑 → (𝐵 · -1) = (-1 · 𝐵)) |
19 | 17 | mulm1d 8172 | . . . 4 ⊢ (𝜑 → (-1 · 𝐵) = -𝐵) |
20 | 18, 19 | eqtrd 2172 | . . 3 ⊢ (𝜑 → (𝐵 · -1) = -𝐵) |
21 | 20 | oveq1d 5789 | . 2 ⊢ (𝜑 → ((𝐵 · -1) mod 𝐶) = (-𝐵 mod 𝐶)) |
22 | 8, 15, 21 | 3eqtr3d 2180 | 1 ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 · cmul 7625 < clt 7800 -cneg 7934 ℤcz 9054 ℚcq 9411 mod cmo 10095 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 df-mod 10096 |
This theorem is referenced by: modqsub12d 10154 |
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