Theorem dmaddpqlem 7185

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7187. (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 6481	. . 3 ⊢ (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2		elxpi 4555	. . . . . . . 8 ⊢ (𝑎 ∈ (N × N) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
3		simpl 108	. . . . . . . . 9 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
4	3	2eximi 1580	. . . . . . . 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 338	. . . . . 6 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7		19.41vv 1875	. . . . . 6 ⊢ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
8	6, 7	sylibr 133	. . . . 5 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9		simpr 109	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10		eceq1 6464	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
11	10	adantr 274	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
12	9, 11	eqtrd 2172	. . . . . 6 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13	12	2eximi 1580	. . . . 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 2544	. . 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 7156	. 2 ⊢ Q = ((N × N) / ~Q )
18	16, 17	eleq2s 2234	1 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  ⟨cop 3530   × cxp 4537  [cec 6427   / cqs 6428  Ncnpi 7080   ~_Q ceq 7087  Qcnq 7088

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435  df-nqqs 7156

This theorem is referenced by:  dmaddpq  7187  dmmulpq  7188