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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 12690 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | zre 12323 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
3 | nnre 11980 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
4 | nnne0 12007 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
5 | 3, 4 | jca 512 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 𝑦 ≠ 0)) |
6 | redivcl 11694 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℝ) | |
7 | 6 | 3expb 1119 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 𝑦 ≠ 0)) → (𝑥 / 𝑦) ∈ ℝ) |
8 | 2, 5, 7 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℝ) |
9 | eleq1 2826 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℝ ↔ (𝑥 / 𝑦) ∈ ℝ)) | |
10 | 8, 9 | syl5ibrcom 246 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ)) |
11 | 10 | rexlimivv 3221 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ) |
12 | 1, 11 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 (class class class)co 7275 ℝcr 10870 0cc0 10871 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℚcq 12688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-z 12320 df-q 12689 |
This theorem is referenced by: qred 12695 qssre 12699 irradd 12713 irrmul 12714 qbtwnxr 12934 qsqueeze 12935 qextltlem 12936 xralrple 12939 ixxub 13100 ixxlb 13101 ioo0 13104 ico0 13125 ioc0 13126 qnumgt0 16454 pcabs 16576 blssps 23577 blss 23578 blcld 23661 qdensere 23933 nmoleub2lem3 24278 mbfaddlem 24824 dvlip2 25159 itgsubst 25213 aalioulem2 25493 aalioulem4 25495 aalioulem5 25496 aalioulem6 25497 aaliou 25498 aaliou2b 25501 ipasslem8 29199 irrdifflemf 35496 itg2gt0cn 35832 3cubeslem1 40506 3cubeslem2 40507 3cubeslem3r 40509 3cubeslem4 40511 irrapxlem5 40648 rpnnen3lem 40853 qinioo 43073 qelioo 43084 qndenserrnbllem 43835 smfaddlem1 44298 smfaddlem2 44299 |
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