Theorem dmaddpqlem 6567

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6569. (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 6181	. . 3 ⊢ (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2		elxpi 4379	. . . . . . . 8 ⊢ (𝑎 ∈ (N × N) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
3		simpl 107	. . . . . . . . 9 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
4	3	2eximi 1532	. . . . . . . 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 333	. . . . . 6 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7		19.41vv 1824	. . . . . 6 ⊢ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤∃𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
8	6, 7	sylibr 132	. . . . 5 ⊢ ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9		simpr 108	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10		eceq1 6164	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
11	10	adantr 270	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
12	9, 11	eqtrd 2113	. . . . . 6 ⊢ ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13	12	2eximi 1532	. . . . 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 2472	. . 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 6538	. 2 ⊢ Q = ((N × N) / ~Q )
18	16, 17	eleq2s 2173	1 ⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  ⟨cop 3401   × cxp 4361  [cec 6127   / cqs 6128  Ncnpi 6462   ~_Q ceq 6469  Qcnq 6470

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131  df-qs 6135  df-nqqs 6538

This theorem is referenced by:  dmaddpq  6569  dmmulpq  6570