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Mirrors > Home > ILE Home > Th. List > q2txmodxeq0 | GIF version |
Description: Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.) |
Ref | Expression |
---|---|
q2txmodxeq0 | ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) mod 𝑋) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cnd 8786 | . . . 4 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 2 ∈ ℂ) | |
2 | qcn 9419 | . . . . 5 ⊢ (𝑋 ∈ ℚ → 𝑋 ∈ ℂ) | |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 ∈ ℂ) |
4 | qre 9410 | . . . . . 6 ⊢ (𝑋 ∈ ℚ → 𝑋 ∈ ℝ) | |
5 | 4 | adantr 274 | . . . . 5 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 ∈ ℝ) |
6 | simpr 109 | . . . . 5 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 0 < 𝑋) | |
7 | 5, 6 | gt0ap0d 8384 | . . . 4 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 # 0) |
8 | 1, 3, 7 | divcanap4d 8549 | . . 3 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) / 𝑋) = 2) |
9 | 2z 9075 | . . 3 ⊢ 2 ∈ ℤ | |
10 | 8, 9 | eqeltrdi 2228 | . 2 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) / 𝑋) ∈ ℤ) |
11 | zq 9411 | . . . . . 6 ⊢ (2 ∈ ℤ → 2 ∈ ℚ) | |
12 | 9, 11 | ax-mp 5 | . . . . 5 ⊢ 2 ∈ ℚ |
13 | qmulcl 9422 | . . . . 5 ⊢ ((2 ∈ ℚ ∧ 𝑋 ∈ ℚ) → (2 · 𝑋) ∈ ℚ) | |
14 | 12, 13 | mpan 420 | . . . 4 ⊢ (𝑋 ∈ ℚ → (2 · 𝑋) ∈ ℚ) |
15 | 14 | adantr 274 | . . 3 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → (2 · 𝑋) ∈ ℚ) |
16 | simpl 108 | . . 3 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 ∈ ℚ) | |
17 | modq0 10095 | . . 3 ⊢ (((2 · 𝑋) ∈ ℚ ∧ 𝑋 ∈ ℚ ∧ 0 < 𝑋) → (((2 · 𝑋) mod 𝑋) = 0 ↔ ((2 · 𝑋) / 𝑋) ∈ ℤ)) | |
18 | 15, 16, 6, 17 | syl3anc 1216 | . 2 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → (((2 · 𝑋) mod 𝑋) = 0 ↔ ((2 · 𝑋) / 𝑋) ∈ ℤ)) |
19 | 10, 18 | mpbird 166 | 1 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) mod 𝑋) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 ℝcr 7612 0cc0 7613 · cmul 7618 < clt 7793 / cdiv 8425 2c2 8764 ℤcz 9047 ℚcq 9404 mod cmo 10088 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-q 9405 df-rp 9435 df-fl 10036 df-mod 10089 |
This theorem is referenced by: (None) |
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