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Theorem qredeu 12051
Description: Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
Assertion
Ref Expression
qredeu (𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem qredeu
Dummy variables 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnz 9231 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2 gcddvds 11918 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
32simpld 111 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
41, 3sylan2 284 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5 gcdcl 11921 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ0)
61, 5sylan2 284 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ0)
76nn0zd 9332 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8 simpl 108 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
91adantl 275 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10 nnne0 8906 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
1110neneqd 2361 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
1211intnand 926 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
1312adantl 275 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14 gcdn0cl 11917 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
158, 9, 13, 14syl21anc 1232 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
1615nnne0d 8923 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
17 dvdsval2 11752 . . . . . . . . . 10 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
187, 16, 8, 17syl3anc 1233 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
194, 18mpbid 146 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
20193adant3 1012 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
212simprd 113 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
221, 21sylan2 284 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
23 dvdsval2 11752 . . . . . . . . . . . 12 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
247, 16, 9, 23syl3anc 1233 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
2522, 24mpbid 146 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
26 nnre 8885 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
2726adantl 275 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
286nn0red 9189 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
29 nngt0 8903 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
3029adantl 275 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
3115nngt0d 8922 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
3227, 28, 30, 31divgt0d 8851 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
3325, 32jca 304 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
34333adant3 1012 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
35 elnnz 9222 . . . . . . . 8 ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
3634, 35sylibr 133 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
37 opelxpi 4643 . . . . . . 7 (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
3820, 36, 37syl2anc 409 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
39 fveq2 5496 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1st𝑥) = (1st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩))
40 simp1 992 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑧 ∈ ℤ)
41153adant3 1012 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 gcd 𝑛) ∈ ℕ)
42 znq 9583 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
4340, 41, 42syl2anc 409 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
4493adant3 1012 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑛 ∈ ℤ)
45 znq 9583 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
4644, 41, 45syl2anc 409 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
47 op1stg 6129 . . . . . . . . . . 11 (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ) → (1st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑧 / (𝑧 gcd 𝑛)))
4843, 46, 47syl2anc 409 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (1st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑧 / (𝑧 gcd 𝑛)))
4939, 48sylan9eqr 2225 . . . . . . . . 9 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (1st𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
50 fveq2 5496 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2nd𝑥) = (2nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩))
51 op2ndg 6130 . . . . . . . . . . 11 (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ) → (2nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑛 / (𝑧 gcd 𝑛)))
5243, 46, 51syl2anc 409 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (2nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑛 / (𝑧 gcd 𝑛)))
5350, 52sylan9eqr 2225 . . . . . . . . 9 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (2nd𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
5449, 53oveq12d 5871 . . . . . . . 8 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → ((1st𝑥) gcd (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))
5554eqeq1d 2179 . . . . . . 7 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1))
5649, 53oveq12d 5871 . . . . . . . 8 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → ((1st𝑥) / (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
5756eqeq2d 2182 . . . . . . 7 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
5855, 57anbi12d 470 . . . . . 6 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))))
5919, 25gcdcld 11923 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ0)
6059nn0cnd 9190 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
61 1cnd 7936 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
626nn0cnd 9190 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
6315nnap0d 8924 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) # 0)
6462mulid1d 7937 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
65 zcn 9217 . . . . . . . . . . . . 13 (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
6665adantr 274 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
6766, 62, 63divcanap2d 8709 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
68 nncn 8886 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
6968adantl 275 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
7069, 62, 63divcanap2d 8709 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
7167, 70oveq12d 5871 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = (𝑧 gcd 𝑛))
72 mulgcd 11971 . . . . . . . . . . 11 (((𝑧 gcd 𝑛) ∈ ℕ0 ∧ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))))
736, 19, 25, 72syl3anc 1233 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))))
7464, 71, 733eqtr2rd 2210 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · 1))
7560, 61, 62, 63, 74mulcanapad 8581 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
76753adant3 1012 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
77 nnap0 8907 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 # 0)
7877adantl 275 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 # 0)
7966, 69, 62, 78, 63divcanap7d 8736 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
8079eqeq2d 2182 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
8180biimp3ar 1341 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
8276, 81jca 304 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
8338, 58, 82rspcedvd 2840 . . . . 5 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
84 elxp6 6148 . . . . . . 7 (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)))
85 elxp6 6148 . . . . . . 7 (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)))
86 simprl 526 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
8786ad2antrr 485 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → (1st𝑥) ∈ ℤ)
88 simprr 527 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
8988ad2antrr 485 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → (2nd𝑥) ∈ ℕ)
90 simprll 532 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → ((1st𝑥) gcd (2nd𝑥)) = 1)
91 simprl 526 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (1st𝑦) ∈ ℤ)
9291ad2antlr 486 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → (1st𝑦) ∈ ℤ)
93 simprr 527 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (2nd𝑦) ∈ ℕ)
9493ad2antlr 486 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → (2nd𝑦) ∈ ℕ)
95 simprrl 534 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → ((1st𝑦) gcd (2nd𝑦)) = 1)
96 simprlr 533 . . . . . . . . . . . 12 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → 𝐴 = ((1st𝑥) / (2nd𝑥)))
97 simprrr 535 . . . . . . . . . . . 12 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → 𝐴 = ((1st𝑦) / (2nd𝑦)))
9896, 97eqtr3d 2205 . . . . . . . . . . 11 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦)))
99 qredeq 12050 . . . . . . . . . . 11 ((((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ ∧ ((1st𝑥) gcd (2nd𝑥)) = 1) ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ ∧ ((1st𝑦) gcd (2nd𝑦)) = 1) ∧ ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦))) → ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
10087, 89, 90, 92, 94, 95, 98, 99syl331anc 1258 . . . . . . . . . 10 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
101 vex 2733 . . . . . . . . . . . 12 𝑥 ∈ V
102 1stexg 6146 . . . . . . . . . . . 12 (𝑥 ∈ V → (1st𝑥) ∈ V)
103101, 102ax-mp 5 . . . . . . . . . . 11 (1st𝑥) ∈ V
104 2ndexg 6147 . . . . . . . . . . . 12 (𝑥 ∈ V → (2nd𝑥) ∈ V)
105101, 104ax-mp 5 . . . . . . . . . . 11 (2nd𝑥) ∈ V
106103, 105opth 4222 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
107100, 106sylibr 133 . . . . . . . . 9 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
108 simplll 528 . . . . . . . . 9 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
109 simplrl 530 . . . . . . . . 9 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
110107, 108, 1093eqtr4d 2213 . . . . . . . 8 ((((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) ∧ ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦))))) → 𝑥 = 𝑦)
111110ex 114 . . . . . . 7 (((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ))) → (((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))
11284, 85, 111syl2anb 289 . . . . . 6 ((𝑥 ∈ (ℤ × ℕ) ∧ 𝑦 ∈ (ℤ × ℕ)) → (((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))
113112rgen2a 2524 . . . . 5 𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦)
11483, 113jctir 311 . . . 4 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦)))
1151143expia 1200 . . 3 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = (𝑧 / 𝑛) → (∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))))
116115rexlimivv 2593 . 2 (∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛) → (∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦)))
117 elq 9581 . 2 (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
118 fveq2 5496 . . . . . 6 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
119 fveq2 5496 . . . . . 6 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
120118, 119oveq12d 5871 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) gcd (2nd𝑥)) = ((1st𝑦) gcd (2nd𝑦)))
121120eqeq1d 2179 . . . 4 (𝑥 = 𝑦 → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((1st𝑦) gcd (2nd𝑦)) = 1))
122118, 119oveq12d 5871 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦)))
123122eqeq2d 2182 . . . 4 (𝑥 = 𝑦 → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑦) / (2nd𝑦))))
124121, 123anbi12d 470 . . 3 (𝑥 = 𝑦 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))))
125124reu4 2924 . 2 (∃!𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦)))
126116, 117, 1253imtr4i 200 1 (𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  wne 2340  wral 2448  wrex 2449  ∃!wreu 2450  Vcvv 2730  cop 3586   class class class wbr 3989   × cxp 4609  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  cc 7772  cr 7773  0cc0 7774  1c1 7775   · cmul 7779   < clt 7954   # cap 8500   / cdiv 8589  cn 8878  0cn0 9135  cz 9212  cq 9578  cdvds 11749   gcd cgcd 11897
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898
This theorem is referenced by:  qnumdencl  12141  qnumdenbi  12146
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