Theorem nqpi 7498

Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7497 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)

Assertion

Ref	Expression
nqpi	⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Distinct variable group:   𝑣,𝐴,𝑤

Proof of Theorem nqpi

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 6681	. . 3 ⊢ (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2		elxpi 4695	. . . . . . 7 ⊢ (𝑎 ∈ (N × N) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
3	2	anim1i 340	. . . . . 6 ⊢ ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ))
4		19.41vv 1928	. . . . . 6 ⊢ (∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) ↔ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ))
5	3, 4	sylibr 134	. . . . 5 ⊢ ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ))
6		simplr 528	. . . . . . 7 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → (𝑤 ∈ N ∧ 𝑣 ∈ N))
7		simpr 110	. . . . . . . 8 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [𝑎] ~Q )
8		eceq1 6662	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
9	8	ad2antrr 488	. . . . . . . 8 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
10	7, 9	eqtrd 2239	. . . . . . 7 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
11	6, 10	jca 306	. . . . . 6 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12	11	2eximi 1625	. . . . 5 ⊢ (∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
13	5, 12	syl 14	. . . 4 ⊢ ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
14	13	rexlimiva 2619	. . 3 ⊢ (∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
15	1, 14	syl 14	. 2 ⊢ (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
16		df-nqqs 7468	. 2 ⊢ Q = ((N × N) / ~Q )
17	15, 16	eleq2s 2301	1 ⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486  ⟨cop 3637   × cxp 4677  [cec 6625   / cqs 6626  Ncnpi 7392   ~_Q ceq 7399  Qcnq 7400

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-cnv 4687  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-ec 6629  df-qs 6633  df-nqqs 7468

This theorem is referenced by:  ltdcnq  7517  archnqq  7537  nqpnq0nq  7573  nqnq0a  7574  nqnq0m  7575