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Mirrors > Home > ILE Home > Th. List > qre | 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 9382 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | zre 9026 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
3 | nnre 8695 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
4 | nnap0 8717 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0) | |
5 | 3, 4 | jca 304 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 𝑦 # 0)) |
6 | redivclap 8459 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑦 # 0) → (𝑥 / 𝑦) ∈ ℝ) | |
7 | 6 | 3expb 1167 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 𝑦 # 0)) → (𝑥 / 𝑦) ∈ ℝ) |
8 | 2, 5, 7 | syl2an 287 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℝ) |
9 | eleq1 2180 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℝ ↔ (𝑥 / 𝑦) ∈ ℝ)) | |
10 | 8, 9 | syl5ibrcom 156 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ)) |
11 | 10 | rexlimivv 2532 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ) |
12 | 1, 11 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 class class class wbr 3899 (class class class)co 5742 ℝcr 7587 0cc0 7588 # cap 8311 / cdiv 8400 ℕcn 8688 ℤcz 9022 ℚcq 9379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-z 9023 df-q 9380 |
This theorem is referenced by: qssre 9390 qltlen 9400 qlttri2 9401 irradd 9406 irrmul 9407 qletric 9989 qlelttric 9990 qltnle 9991 qdceq 9992 qbtwnz 9997 qbtwnxr 10003 qavgle 10004 ioo0 10005 ioom 10006 ico0 10007 ioc0 10008 flqcl 10014 flqlelt 10017 qfraclt1 10021 qfracge0 10022 flqge 10023 flqltnz 10028 flqwordi 10029 flqbi 10031 flqbi2 10032 flqaddz 10038 flqmulnn0 10040 flltdivnn0lt 10045 ceilqval 10047 ceiqge 10050 ceiqm1l 10052 ceiqle 10054 flqleceil 10058 flqeqceilz 10059 intfracq 10061 flqdiv 10062 modqval 10065 modq0 10070 mulqmod0 10071 negqmod0 10072 modqge0 10073 modqlt 10074 modqelico 10075 modqdiffl 10076 modqmulnn 10083 modqid 10090 modqid0 10091 modqabs 10098 modqabs2 10099 modqcyc 10100 mulqaddmodid 10105 modqmuladdim 10108 modqmuladdnn0 10109 modqltm1p1mod 10117 q2txmodxeq0 10125 q2submod 10126 modqdi 10133 modqsubdir 10134 fimaxq 10541 qabsor 10815 qdenre 10942 expcnvre 11240 flodddiv4t2lthalf 11561 sqrt2irraplemnn 11784 sqrt2irrap 11785 qnumgt0 11803 blssps 12523 blss 12524 qtopbas 12618 qdencn 13149 |
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