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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 9021 | . . . 4 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 2 ∈ ℂ) | |
2 | qcn 9663 | . . . . 5 ⊢ (𝑋 ∈ ℚ → 𝑋 ∈ ℂ) | |
3 | 2 | adantr 276 | . . . 4 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 ∈ ℂ) |
4 | qre 9654 | . . . . . 6 ⊢ (𝑋 ∈ ℚ → 𝑋 ∈ ℝ) | |
5 | 4 | adantr 276 | . . . . 5 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 ∈ ℝ) |
6 | simpr 110 | . . . . 5 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 0 < 𝑋) | |
7 | 5, 6 | gt0ap0d 8615 | . . . 4 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 # 0) |
8 | 1, 3, 7 | divcanap4d 8782 | . . 3 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) / 𝑋) = 2) |
9 | 2z 9310 | . . 3 ⊢ 2 ∈ ℤ | |
10 | 8, 9 | eqeltrdi 2280 | . 2 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) / 𝑋) ∈ ℤ) |
11 | zq 9655 | . . . . . 6 ⊢ (2 ∈ ℤ → 2 ∈ ℚ) | |
12 | 9, 11 | ax-mp 5 | . . . . 5 ⊢ 2 ∈ ℚ |
13 | qmulcl 9666 | . . . . 5 ⊢ ((2 ∈ ℚ ∧ 𝑋 ∈ ℚ) → (2 · 𝑋) ∈ ℚ) | |
14 | 12, 13 | mpan 424 | . . . 4 ⊢ (𝑋 ∈ ℚ → (2 · 𝑋) ∈ ℚ) |
15 | 14 | adantr 276 | . . 3 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → (2 · 𝑋) ∈ ℚ) |
16 | simpl 109 | . . 3 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → 𝑋 ∈ ℚ) | |
17 | modq0 10359 | . . 3 ⊢ (((2 · 𝑋) ∈ ℚ ∧ 𝑋 ∈ ℚ ∧ 0 < 𝑋) → (((2 · 𝑋) mod 𝑋) = 0 ↔ ((2 · 𝑋) / 𝑋) ∈ ℤ)) | |
18 | 15, 16, 6, 17 | syl3anc 1249 | . 2 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → (((2 · 𝑋) mod 𝑋) = 0 ↔ ((2 · 𝑋) / 𝑋) ∈ ℤ)) |
19 | 10, 18 | mpbird 167 | 1 ⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) mod 𝑋) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5895 ℂcc 7838 ℝcr 7839 0cc0 7840 · cmul 7845 < clt 8021 / cdiv 8658 2c2 8999 ℤcz 9282 ℚcq 9648 mod cmo 10352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-n0 9206 df-z 9283 df-q 9649 df-rp 9683 df-fl 10300 df-mod 10353 |
This theorem is referenced by: (None) |
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