Theorem dmaddpqlem 7734

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7736. (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 6851	. . 3 ⊢ (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2		elxpi 4785	. . . . . . . 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 1654	. . . . . . . 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 1959	. . . . . 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 6832	. . . . . . . 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 2271	. . . . . 6 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13	12	2eximi 1654	. . . . 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 2663	. . 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 7705	. 2 ⊢ Q = ((N × N) / ~Q )
18	16, 17	eleq2s 2333	1 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3708   × cxp 4767  [cec 6795   / cqs 6796  Ncnpi 7629   ~_Q ceq 7636  Qcnq 7637

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-ec 6799  df-qs 6803  df-nqqs 7705

This theorem is referenced by:  dmaddpq  7736  dmmulpq  7737