Theorem nqpi 7391

Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7390 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 6601	. . 3 ⊢ (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2		elxpi 4654	. . . . . . 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 1913	. . . . . 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 6584	. . . . . . . . 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 2220	. . . . . . 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 1611	. . . . 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 2599	. . 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 7361	. 2 ⊢ Q = ((N × N) / ~Q )
17	15, 16	eleq2s 2282	1 ⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363  ∃wex 1502   ∈ wcel 2158  ∃wrex 2466  ⟨cop 3607   × cxp 4636  [cec 6547   / cqs 6548  Ncnpi 7285   ~_Q ceq 7292  Qcnq 7293

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-ec 6551  df-qs 6555  df-nqqs 7361

This theorem is referenced by:  ltdcnq  7410  archnqq  7430  nqpnq0nq  7466  nqnq0a  7467  nqnq0m  7468