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Mirrors > Home > ILE Home > Th. List > recnnre | GIF version |
Description: Embedding the reciprocal of a natural number into ℝ. (Contributed by Jim Kingdon, 15-Jul-2021.) |
Ref | Expression |
---|---|
recnnre | ⊢ (𝑁 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recnnpr 6646 | . . . . . 6 ⊢ (𝑁 ∈ N → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 ∈ P) | |
2 | 1pr 6652 | . . . . . 6 ⊢ 1P ∈ P | |
3 | addclpr 6635 | . . . . . 6 ⊢ ((〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 ∈ P ∧ 1P ∈ P) → (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P) ∈ P) | |
4 | 1, 2, 3 | sylancl 392 | . . . . 5 ⊢ (𝑁 ∈ N → (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P) ∈ P) |
5 | opelxpi 4376 | . . . . 5 ⊢ (((〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P) ∈ P ∧ 1P ∈ P) → 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 ∈ (P × P)) | |
6 | 4, 2, 5 | sylancl 392 | . . . 4 ⊢ (𝑁 ∈ N → 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 ∈ (P × P)) |
7 | enrex 6822 | . . . . 5 ⊢ ~R ∈ V | |
8 | 7 | ecelqsi 6160 | . . . 4 ⊢ (〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 ∈ (P × P) → [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ∈ ((P × P) / ~R )) |
9 | 6, 8 | syl 14 | . . 3 ⊢ (𝑁 ∈ N → [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ∈ ((P × P) / ~R )) |
10 | df-nr 6812 | . . 3 ⊢ R = ((P × P) / ~R ) | |
11 | 9, 10 | syl6eleqr 2131 | . 2 ⊢ (𝑁 ∈ N → [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ∈ R) |
12 | opelreal 6904 | . 2 ⊢ (〈[〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ℝ ↔ [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ∈ R) | |
13 | 11, 12 | sylibr 137 | 1 ⊢ (𝑁 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1𝑜〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1𝑜〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 {cab 2026 〈cop 3378 class class class wbr 3764 × cxp 4343 ‘cfv 4902 (class class class)co 5512 1𝑜c1o 5994 [cec 6104 / cqs 6105 Ncnpi 6370 ~Q ceq 6377 *Qcrq 6382 <Q cltq 6383 Pcnp 6389 1Pc1p 6390 +P cpp 6391 ~R cer 6394 Rcnr 6395 0Rc0r 6396 ℝcr 6888 |
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-2o 6002 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-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-enr 6811 df-nr 6812 df-0r 6816 df-r 6899 |
This theorem is referenced by: recriota 6964 |
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