Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7480 | . . . . . 6 ⊢ (𝑁 ∈ N → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉 ∈ P) | |
2 | 1pr 7486 | . . . . . 6 ⊢ 1P ∈ P | |
3 | addclpr 7469 | . . . . . 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 410 | . . . . 5 ⊢ (𝑁 ∈ N → (〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P) ∈ P) |
5 | opelxpi 4630 | . . . . 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 410 | . . . 4 ⊢ (𝑁 ∈ N → 〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉 ∈ (P × P)) |
7 | enrex 7669 | . . . . 5 ⊢ ~R ∈ V | |
8 | 7 | ecelqsi 6546 | . . . 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 7659 | . . 3 ⊢ R = ((P × P) / ~R ) | |
11 | 9, 10 | eleqtrrdi 2258 | . 2 ⊢ (𝑁 ∈ N → [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ∈ R) |
12 | opelreal 7759 | . 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 133 | 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 2135 {cab 2150 〈cop 3573 class class class wbr 3976 × cxp 4596 ‘cfv 5182 (class class class)co 5836 1oc1o 6368 [cec 6490 / cqs 6491 Ncnpi 7204 ~Q ceq 7211 *Qcrq 7216 <Q cltq 7217 Pcnp 7223 1Pc1p 7224 +P cpp 7225 ~R cer 7228 Rcnr 7229 0Rc0r 7230 ℝcr 7743 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-enr 7658 df-nr 7659 df-0r 7663 df-r 7754 |
This theorem is referenced by: recriota 7822 |
Copyright terms: Public domain | W3C validator |