nnnq - Intuitionistic Logic Explorer

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Theorem nnnq 7685

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 7578	. . . 4 ⊢ 1_o ∈ N
2		opelxpi 4763	. . . 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 7623	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6801	. . 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 7611	. 2 ⊢ Q = ((N × N) / ~_Q )
8	6, 7	eleqtrrdi 2325	1 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2202 ⟨cop 3676 × cxp 4729 1_oc1o 6618 [cec 6743 / cqs 6744 Ncnpi 7535 ~_Q ceq 7542 Qcnq 7543

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-br 4094 df-opab 4156 df-suc 4474 df-iom 4695 df-xp 4737 df-cnv 4739 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-1o 6625 df-ec 6747 df-qs 6751 df-ni 7567 df-enq 7610 df-nqqs 7611