Theorem nqpi 7462

Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7461 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 6655	. . 3 ⊢ (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2		elxpi 4680	. . . . . . 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 1918	. . . . . 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 6636	. . . . . . . . 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 2229	. . . . . . 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 1615	. . . . 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 2609	. . 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 7432	. 2 ⊢ Q = ((N × N) / ~Q )
17	15, 16	eleq2s 2291	1 ⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  ⟨cop 3626   × cxp 4662  [cec 6599   / cqs 6600  Ncnpi 7356   ~_Q ceq 7363  Qcnq 7364

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-ec 6603  df-qs 6607  df-nqqs 7432

This theorem is referenced by:  ltdcnq  7481  archnqq  7501  nqpnq0nq  7537  nqnq0a  7538  nqnq0m  7539