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Mirrors > Home > MPE Home > Th. List > qmulz | Structured version Visualization version GIF version |
Description: If 𝐴 is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
qmulz | ⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 12349 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℕ 𝐴 = (𝑦 / 𝑥)) | |
2 | rexcom 3355 | . . 3 ⊢ (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℕ 𝐴 = (𝑦 / 𝑥) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℤ 𝐴 = (𝑦 / 𝑥)) | |
3 | zcn 11985 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
4 | 3 | adantl 484 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ) |
5 | nncn 11645 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
6 | 5 | adantr 483 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ) |
7 | nnne0 11670 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0) | |
8 | 7 | adantr 483 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → 𝑥 ≠ 0) |
9 | 4, 6, 8 | divcan1d 11416 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → ((𝑦 / 𝑥) · 𝑥) = 𝑦) |
10 | simpr 487 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ) | |
11 | 9, 10 | eqeltrd 2913 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → ((𝑦 / 𝑥) · 𝑥) ∈ ℤ) |
12 | oveq1 7162 | . . . . . . 7 ⊢ (𝐴 = (𝑦 / 𝑥) → (𝐴 · 𝑥) = ((𝑦 / 𝑥) · 𝑥)) | |
13 | 12 | eleq1d 2897 | . . . . . 6 ⊢ (𝐴 = (𝑦 / 𝑥) → ((𝐴 · 𝑥) ∈ ℤ ↔ ((𝑦 / 𝑥) · 𝑥) ∈ ℤ)) |
14 | 11, 13 | syl5ibrcom 249 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℤ) → (𝐴 = (𝑦 / 𝑥) → (𝐴 · 𝑥) ∈ ℤ)) |
15 | 14 | rexlimdva 3284 | . . . 4 ⊢ (𝑥 ∈ ℕ → (∃𝑦 ∈ ℤ 𝐴 = (𝑦 / 𝑥) → (𝐴 · 𝑥) ∈ ℤ)) |
16 | 15 | reximia 3242 | . . 3 ⊢ (∃𝑥 ∈ ℕ ∃𝑦 ∈ ℤ 𝐴 = (𝑦 / 𝑥) → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) |
17 | 2, 16 | sylbi 219 | . 2 ⊢ (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℕ 𝐴 = (𝑦 / 𝑥) → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) |
18 | 1, 17 | sylbi 219 | 1 ⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 (class class class)co 7155 ℂcc 10534 0cc0 10536 · cmul 10541 / cdiv 11296 ℕcn 11637 ℤcz 11980 ℚcq 12347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-z 11981 df-q 12348 |
This theorem is referenced by: elqaalem1 24907 elqaalem3 24909 |
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