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Mirrors > Home > ILE Home > Th. List > prarloclem | GIF version |
Description: A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from 𝐴 to 𝐴 +Q (𝑁 ·Q 𝑃) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.) |
Ref | Expression |
---|---|
prarloclem | ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([〈𝑁, 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prarloclem5 7332 | . 2 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([〈𝑁, 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) | |
2 | prarloclem4 7330 | . . . 4 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ 𝑃 ∈ Q) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) | |
3 | 2 | 3ad2antr2 1148 | . . 3 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁)) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
4 | 3 | 3adant3 1002 | . 2 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([〈𝑁, 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
5 | 1, 4 | mpd 13 | 1 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([〈𝑁, 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 1481 ∃wrex 2418 〈cop 3535 class class class wbr 3937 ωcom 4512 (class class class)co 5782 1oc1o 6314 2oc2o 6315 +o coa 6318 [cec 6435 Ncnpi 7104 <N clti 7107 ~Q ceq 7111 Qcnq 7112 +Q cplq 7114 ·Q cmq 7115 ~Q0 ceq0 7118 +Q0 cplq0 7121 ·Q0 cmq0 7122 Pcnp 7123 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 |
This theorem is referenced by: prarloc 7335 |
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