Theorem dmaddpqlem 7640

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7642. (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 6799	. . 3 ⊢ (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2		elxpi 4747	. . . . . . . 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 1650	. . . . . . . 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 1952	. . . . . 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 6780	. . . . . . . 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 2264	. . . . . 6 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13	12	2eximi 1650	. . . . 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 2646	. . 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 7611	. 2 ⊢ Q = ((N × N) / ~Q )
18	16, 17	eleq2s 2326	1 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∃wrex 2512  ⟨cop 3676   × cxp 4729  [cec 6743   / cqs 6744  Ncnpi 7535   ~_Q ceq 7542  Qcnq 7543

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-ec 6747  df-qs 6751  df-nqqs 7611

This theorem is referenced by:  dmaddpq  7642  dmmulpq  7643