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Mirrors > Home > ILE Home > Th. List > prarloc2 | GIF version |
Description: A Dedekind cut is arithmetically located. This is a variation of prarloc 7533 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.) |
Ref | Expression |
---|---|
prarloc2 | ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 (𝑎 +Q 𝑃) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prarloc 7533 | . 2 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃)) | |
2 | prcunqu 7515 | . . . . 5 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝑏 ∈ 𝑈) → (𝑏 <Q (𝑎 +Q 𝑃) → (𝑎 +Q 𝑃) ∈ 𝑈)) | |
3 | 2 | rexlimdva 2607 | . . . 4 ⊢ (〈𝐿, 𝑈〉 ∈ P → (∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃) → (𝑎 +Q 𝑃) ∈ 𝑈)) |
4 | 3 | reximdv 2591 | . . 3 ⊢ (〈𝐿, 𝑈〉 ∈ P → (∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃) → ∃𝑎 ∈ 𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)) |
5 | 4 | adantr 276 | . 2 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝑃 ∈ Q) → (∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃) → ∃𝑎 ∈ 𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)) |
6 | 1, 5 | mpd 13 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 (𝑎 +Q 𝑃) ∈ 𝑈) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 ∃wrex 2469 〈cop 3610 class class class wbr 4018 (class class class)co 5897 Qcnq 7310 +Q cplq 7312 <Q cltq 7315 Pcnp 7321 |
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 |
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-ral 2473 df-rex 2474 df-reu 2475 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-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-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 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-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-1o 6442 df-2o 6443 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-pli 7335 df-mi 7336 df-lti 7337 df-plpq 7374 df-mpq 7375 df-enq 7377 df-nqqs 7378 df-plqqs 7379 df-mqqs 7380 df-1nqqs 7381 df-rq 7382 df-ltnqqs 7383 df-enq0 7454 df-nq0 7455 df-0nq0 7456 df-plq0 7457 df-mq0 7458 df-inp 7496 |
This theorem is referenced by: addcanprleml 7644 addcanprlemu 7645 aptiprleml 7669 aptiprlemu 7670 |
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