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Mirrors > Home > MPE Home > Th. List > elpqb | Structured version Visualization version GIF version |
Description: A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.) |
Ref | Expression |
---|---|
elpqb | ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpq 12377 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | nnz 12007 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ) | |
3 | znq 12355 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℚ) | |
4 | 2, 3 | sylan 582 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℚ) |
5 | nnre 11647 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ) | |
6 | nngt0 11671 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → 0 < 𝑥) | |
7 | 5, 6 | jca 514 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → (𝑥 ∈ ℝ ∧ 0 < 𝑥)) |
8 | nnre 11647 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
9 | nngt0 11671 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦) | |
10 | 8, 9 | jca 514 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 0 < 𝑦)) |
11 | divgt0 11510 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 < (𝑥 / 𝑦)) | |
12 | 7, 10, 11 | syl2an 597 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 0 < (𝑥 / 𝑦)) |
13 | 4, 12 | jca 514 | . . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑥 / 𝑦) ∈ ℚ ∧ 0 < (𝑥 / 𝑦))) |
14 | eleq1 2902 | . . . . 5 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℚ ↔ (𝑥 / 𝑦) ∈ ℚ)) | |
15 | breq2 5072 | . . . . 5 ⊢ (𝐴 = (𝑥 / 𝑦) → (0 < 𝐴 ↔ 0 < (𝑥 / 𝑦))) | |
16 | 14, 15 | anbi12d 632 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → ((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ((𝑥 / 𝑦) ∈ ℚ ∧ 0 < (𝑥 / 𝑦)))) |
17 | 13, 16 | syl5ibrcom 249 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℚ ∧ 0 < 𝐴))) |
18 | 17 | rexlimivv 3294 | . 2 ⊢ (∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℚ ∧ 0 < 𝐴)) |
19 | 1, 18 | impbii 211 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 < clt 10677 / cdiv 11299 ℕcn 11640 ℤcz 11984 ℚcq 12351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-z 11985 df-q 12352 |
This theorem is referenced by: (None) |
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