Theorem dmaddpqlem 7472

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7474. (Contributed by Jim Kingdon, 15-Sep-2019.)

Assertion

Ref	Expression
dmaddpqlem	⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Distinct variable group:   𝑤,𝑣,𝑥

Proof of Theorem dmaddpqlem

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 6664	. . 3 ⊢ (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2		elxpi 4689	. . . . . . . 8 ⊢ (𝑎 ∈ (N × N) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
3		simpl 109	. . . . . . . . 9 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
4	3	2eximi 1623	. . . . . . . 8 ⊢ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
5	2, 4	syl 14	. . . . . . 7 ⊢ (𝑎 ∈ (N × N) → ∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
6	5	anim1i 340	. . . . . 6 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7		19.41vv 1926	. . . . . 6 ⊢ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
8	6, 7	sylibr 134	. . . . 5 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9		simpr 110	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10		eceq1 6645	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
11	10	adantr 276	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
12	9, 11	eqtrd 2237	. . . . . 6 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13	12	2eximi 1623	. . . . 5 ⊢ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
14	8, 13	syl 14	. . . 4 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
15	14	rexlimiva 2617	. . 3 ⊢ (∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
16	1, 15	syl 14	. 2 ⊢ (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
17		df-nqqs 7443	. 2 ⊢ Q = ((N × N) / ~Q )
18	16, 17	eleq2s 2299	1 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∃wrex 2484  ⟨cop 3635   × cxp 4671  [cec 6608   / cqs 6609  Ncnpi 7367   ~_Q ceq 7374  Qcnq 7375

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4679  df-cnv 4681  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-ec 6612  df-qs 6616  df-nqqs 7443

This theorem is referenced by:  dmaddpq  7474  dmmulpq  7475