Theorem nqpi 7210

Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7209 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 6489	. . 3 ⊢ (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2		elxpi 4563	. . . . . . 7 ⊢ (𝑎 ∈ (N × N) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
3	2	anim1i 338	. . . . . 6 ⊢ ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ))
4		19.41vv 1876	. . . . . 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 133	. . . . 5 ⊢ ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ))
6		simplr 520	. . . . . . 7 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → (𝑤 ∈ N ∧ 𝑣 ∈ N))
7		simpr 109	. . . . . . . 8 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [𝑎] ~Q )
8		eceq1 6472	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
9	8	ad2antrr 480	. . . . . . . 8 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
10	7, 9	eqtrd 2173	. . . . . . 7 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
11	6, 10	jca 304	. . . . . 6 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12	11	2eximi 1581	. . . . 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 2547	. . 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 7180	. 2 ⊢ Q = ((N × N) / ~Q )
17	15, 16	eleq2s 2235	1 ⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃wrex 2418  ⟨cop 3535   × cxp 4545  [cec 6435   / cqs 6436  Ncnpi 7104   ~_Q ceq 7111  Qcnq 7112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-ec 6439  df-qs 6443  df-nqqs 7180

This theorem is referenced by:  ltdcnq  7229  archnqq  7249  nqpnq0nq  7285  nqnq0a  7286  nqnq0m  7287