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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 7349 | . . . 4 ⊢ 1o ∈ N | |
2 | opelxpi 4679 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → 〈𝐴, 1o〉 ∈ (N × N)) | |
3 | 1, 2 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ N → 〈𝐴, 1o〉 ∈ (N × N)) |
4 | enqex 7394 | . . . 4 ⊢ ~Q ∈ V | |
5 | 4 | ecelqsi 6619 | . . 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 7382 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
8 | 6, 7 | eleqtrrdi 2283 | 1 ⊢ (𝐴 ∈ N → [〈𝐴, 1o〉] ~Q ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 〈cop 3613 × cxp 4645 1oc1o 6438 [cec 6561 / cqs 6562 Ncnpi 7306 ~Q ceq 7313 Qcnq 7314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-iinf 4608 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-suc 4392 df-iom 4611 df-xp 4653 df-cnv 4655 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-1o 6445 df-ec 6565 df-qs 6569 df-ni 7338 df-enq 7381 df-nqqs 7382 |
This theorem is referenced by: recnnpr 7582 nnprlu 7587 archrecnq 7697 archrecpr 7698 caucvgprlemnkj 7700 caucvgprlemnbj 7701 caucvgprlemm 7702 caucvgprlemopl 7703 caucvgprlemlol 7704 caucvgprlemloc 7709 caucvgprlemladdfu 7711 caucvgprlemladdrl 7712 caucvgprprlemloccalc 7718 caucvgprprlemnkltj 7723 caucvgprprlemnkeqj 7724 caucvgprprlemnjltk 7725 caucvgprprlemml 7728 caucvgprprlemopl 7731 caucvgprprlemlol 7732 caucvgprprlemloc 7737 caucvgprprlemexb 7741 caucvgprprlem1 7743 caucvgprprlem2 7744 pitonnlem2 7881 ltrennb 7888 recidpipr 7890 |
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