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Mirrors > Home > ILE Home > Th. List > nnnq | GIF version |
Description: The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Ref | Expression |
---|---|
nnnq | ⊢ (𝐴 ∈ N → [〈𝐴, 1o〉] ~Q ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pi 7091 | . . . 4 ⊢ 1o ∈ N | |
2 | opelxpi 4541 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → 〈𝐴, 1o〉 ∈ (N × N)) | |
3 | 1, 2 | mpan2 421 | . . 3 ⊢ (𝐴 ∈ N → 〈𝐴, 1o〉 ∈ (N × N)) |
4 | enqex 7136 | . . . 4 ⊢ ~Q ∈ V | |
5 | 4 | ecelqsi 6451 | . . 3 ⊢ (〈𝐴, 1o〉 ∈ (N × N) → [〈𝐴, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
6 | 3, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ N → [〈𝐴, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
7 | df-nqqs 7124 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
8 | 6, 7 | eleqtrrdi 2211 | 1 ⊢ (𝐴 ∈ N → [〈𝐴, 1o〉] ~Q ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 〈cop 3500 × cxp 4507 1oc1o 6274 [cec 6395 / cqs 6396 Ncnpi 7048 ~Q ceq 7055 Qcnq 7056 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-suc 4263 df-iom 4475 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-1o 6281 df-ec 6399 df-qs 6403 df-ni 7080 df-enq 7123 df-nqqs 7124 |
This theorem is referenced by: recnnpr 7324 nnprlu 7329 archrecnq 7439 archrecpr 7440 caucvgprlemnkj 7442 caucvgprlemnbj 7443 caucvgprlemm 7444 caucvgprlemopl 7445 caucvgprlemlol 7446 caucvgprlemloc 7451 caucvgprlemladdfu 7453 caucvgprlemladdrl 7454 caucvgprprlemloccalc 7460 caucvgprprlemnkltj 7465 caucvgprprlemnkeqj 7466 caucvgprprlemnjltk 7467 caucvgprprlemml 7470 caucvgprprlemopl 7473 caucvgprprlemlol 7474 caucvgprprlemloc 7479 caucvgprprlemexb 7483 caucvgprprlem1 7485 caucvgprprlem2 7486 pitonnlem2 7623 ltrennb 7630 recidpipr 7632 |
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