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Mirrors > Home > ILE Home > Th. List > nqprxx | GIF version |
Description: The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Ref | Expression |
---|---|
nqprxx | ⊢ (𝐴 ∈ Q → 〈{𝑥 ∣ 𝑥 <Q 𝐴}, {𝑥 ∣ 𝐴 <Q 𝑥}〉 ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqprm 6640 | . . 3 ⊢ (𝐴 ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) | |
2 | ltrelnq 6463 | . . . . . . 7 ⊢ <Q ⊆ (Q × Q) | |
3 | 2 | brel 4392 | . . . . . 6 ⊢ (𝑥 <Q 𝐴 → (𝑥 ∈ Q ∧ 𝐴 ∈ Q)) |
4 | 3 | simpld 105 | . . . . 5 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ Q) |
5 | 4 | abssi 3015 | . . . 4 ⊢ {𝑥 ∣ 𝑥 <Q 𝐴} ⊆ Q |
6 | 2 | brel 4392 | . . . . . 6 ⊢ (𝐴 <Q 𝑥 → (𝐴 ∈ Q ∧ 𝑥 ∈ Q)) |
7 | 6 | simprd 107 | . . . . 5 ⊢ (𝐴 <Q 𝑥 → 𝑥 ∈ Q) |
8 | 7 | abssi 3015 | . . . 4 ⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ⊆ Q |
9 | 5, 8 | pm3.2i 257 | . . 3 ⊢ ({𝑥 ∣ 𝑥 <Q 𝐴} ⊆ Q ∧ {𝑥 ∣ 𝐴 <Q 𝑥} ⊆ Q) |
10 | 1, 9 | jctil 295 | . 2 ⊢ (𝐴 ∈ Q → (({𝑥 ∣ 𝑥 <Q 𝐴} ⊆ Q ∧ {𝑥 ∣ 𝐴 <Q 𝑥} ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))) |
11 | nqprrnd 6641 | . . 3 ⊢ (𝐴 ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})))) | |
12 | nqprdisj 6642 | . . 3 ⊢ (𝐴 ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) | |
13 | nqprloc 6643 | . . 3 ⊢ (𝐴 ∈ Q → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))) | |
14 | 11, 12, 13 | 3jca 1084 | . 2 ⊢ (𝐴 ∈ Q → ((∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})))) |
15 | elinp 6572 | . 2 ⊢ (〈{𝑥 ∣ 𝑥 <Q 𝐴}, {𝑥 ∣ 𝐴 <Q 𝑥}〉 ∈ P ↔ ((({𝑥 ∣ 𝑥 <Q 𝐴} ⊆ Q ∧ {𝑥 ∣ 𝐴 <Q 𝑥} ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))))) | |
16 | 10, 14, 15 | sylanbrc 394 | 1 ⊢ (𝐴 ∈ Q → 〈{𝑥 ∣ 𝑥 <Q 𝐴}, {𝑥 ∣ 𝐴 <Q 𝑥}〉 ∈ P) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 ∧ w3a 885 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∃wrex 2307 ⊆ wss 2917 〈cop 3378 class class class wbr 3764 Qcnq 6378 <Q cltq 6383 Pcnp 6389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 |
This theorem is referenced by: nqprlu 6645 |
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