nnnq - Intuitionistic Logic Explorer

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Theorem nnnq 7456

Description: The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)

**Assertion**
Ref	Expression
nnnq	⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

**Proof of Theorem nnnq**
Step	Hyp	Ref	Expression
1		1pi 7349	. . . 4 ⊢ 1_o ∈ N
2		opelxpi 4679	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1_o ∈ N) → ⟨𝐴, 1_o⟩ ∈ (N × N))
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1_o⟩ ∈ (N × N))
4		enqex 7394	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6619	. . 3 ⊢ (⟨𝐴, 1_o⟩ ∈ (N × N) → [⟨𝐴, 1_o⟩] ~_Q ∈ ((N × N) / ~_Q ))
6	3, 5	syl 14	. 2 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ ((N × N) / ~_Q ))
7		df-nqqs 7382	. 2 ⊢ Q = ((N × N) / ~_Q )
8	6, 7	eleqtrrdi 2283	1 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2160 ⟨cop 3613 × cxp 4645 1_oc1o 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