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Mirrors > Home > ILE Home > Th. List > qdenre | GIF version |
Description: The rational numbers are dense in ℝ: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 10287. (Contributed by BJ, 15-Oct-2021.) |
Ref | Expression |
---|---|
qdenre | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 ∈ ℝ) | |
2 | rpre 9690 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
3 | 2 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ) |
4 | 1, 3 | resubcld 8368 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) ∈ ℝ) |
5 | 1, 3 | readdcld 8017 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 + 𝐵) ∈ ℝ) |
6 | ltsubrp 9720 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴) | |
7 | ltaddrp 9721 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵)) | |
8 | 4, 1, 5, 6, 7 | lttrd 8113 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < (𝐴 + 𝐵)) |
9 | 4, 5, 8 | 3jca 1179 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − 𝐵) ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (𝐴 − 𝐵) < (𝐴 + 𝐵))) |
10 | qbtwnre 10287 | . . 3 ⊢ (((𝐴 − 𝐵) ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (𝐴 − 𝐵) < (𝐴 + 𝐵)) → ∃𝑥 ∈ ℚ ((𝐴 − 𝐵) < 𝑥 ∧ 𝑥 < (𝐴 + 𝐵))) | |
11 | qre 9655 | . . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ) | |
12 | 11 | adantl 277 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ 𝑥 ∈ ℚ) → 𝑥 ∈ ℝ) |
13 | simpll 527 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ 𝑥 ∈ ℚ) → 𝐴 ∈ ℝ) | |
14 | 2 | ad2antlr 489 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ 𝑥 ∈ ℚ) → 𝐵 ∈ ℝ) |
15 | absdiflt 11133 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘(𝑥 − 𝐴)) < 𝐵 ↔ ((𝐴 − 𝐵) < 𝑥 ∧ 𝑥 < (𝐴 + 𝐵)))) | |
16 | 15 | biimprd 158 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 − 𝐵) < 𝑥 ∧ 𝑥 < (𝐴 + 𝐵)) → (abs‘(𝑥 − 𝐴)) < 𝐵)) |
17 | 12, 13, 14, 16 | syl3anc 1249 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ 𝑥 ∈ ℚ) → (((𝐴 − 𝐵) < 𝑥 ∧ 𝑥 < (𝐴 + 𝐵)) → (abs‘(𝑥 − 𝐴)) < 𝐵)) |
18 | 17 | reximdva 2592 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (∃𝑥 ∈ ℚ ((𝐴 − 𝐵) < 𝑥 ∧ 𝑥 < (𝐴 + 𝐵)) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵)) |
19 | 10, 18 | syl5 32 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (((𝐴 − 𝐵) ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (𝐴 − 𝐵) < (𝐴 + 𝐵)) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵)) |
20 | 9, 19 | mpd 13 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 ∈ wcel 2160 ∃wrex 2469 class class class wbr 4018 ‘cfv 5235 (class class class)co 5896 ℝcr 7840 + caddc 7844 < clt 8022 − cmin 8158 ℚcq 9649 ℝ+crp 9683 abscabs 11038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-frec 6416 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-seqfrec 10477 df-exp 10551 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 |
This theorem is referenced by: qdencn 15234 |
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