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Mirrors > Home > MPE Home > Th. List > qre | Structured version Visualization version GIF version |
Description: A rational number is a real number. (Contributed by NM, 14-Nov-2002.) |
Ref | Expression |
---|---|
qre | ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 12353 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | zre 11988 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
3 | nnre 11647 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
4 | nnne0 11674 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
5 | 3, 4 | jca 514 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 𝑦 ≠ 0)) |
6 | redivcl 11361 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℝ) | |
7 | 6 | 3expb 1116 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 𝑦 ≠ 0)) → (𝑥 / 𝑦) ∈ ℝ) |
8 | 2, 5, 7 | syl2an 597 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℝ) |
9 | eleq1 2902 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℝ ↔ (𝑥 / 𝑦) ∈ ℝ)) | |
10 | 8, 9 | syl5ibrcom 249 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ)) |
11 | 10 | rexlimivv 3294 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ) |
12 | 1, 11 | sylbi 219 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 (class class class)co 7158 ℝcr 10538 0cc0 10539 / 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: qssre 12361 irradd 12375 irrmul 12376 qbtwnxr 12596 qsqueeze 12597 qextltlem 12598 xralrple 12601 ixxub 12762 ixxlb 12763 ioo0 12766 ico0 12787 ioc0 12788 qnumgt0 16092 pcabs 16213 blssps 23036 blss 23037 blcld 23117 qdensere 23380 nmoleub2lem3 23721 mbfaddlem 24263 dvlip2 24594 itgsubst 24648 aalioulem2 24924 aalioulem4 24926 aalioulem5 24927 aalioulem6 24928 aaliou 24929 aaliou2b 24932 ipasslem8 28616 itg2gt0cn 34949 3cubeslem1 39288 3cubeslem2 39289 3cubeslem3r 39291 3cubeslem4 39293 irrapxlem5 39430 rpnnen3lem 39635 qred 41664 qinioo 41818 qelioo 41829 qndenserrnbllem 42586 smfaddlem1 43046 smfaddlem2 43047 |
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