Theorem nqpi 7573

Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7572 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 6742	. . 3 ⊢ (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2		elxpi 4735	. . . . . . 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 1950	. . . . . 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 6723	. . . . . . . . 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 2262	. . . . . . 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 1647	. . . . 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 2643	. . 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 7543	. 2 ⊢ Q = ((N × N) / ~Q )
17	15, 16	eleq2s 2324	1 ⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  ⟨cop 3669   × cxp 4717  [cec 6686   / cqs 6687  Ncnpi 7467   ~_Q ceq 7474  Qcnq 7475

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6690  df-qs 6694  df-nqqs 7543

This theorem is referenced by:  ltdcnq  7592  archnqq  7612  nqpnq0nq  7648  nqnq0a  7649  nqnq0m  7650