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Theorem qredeu 11685
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 9027 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2 gcddvds 11559 . . . . . . . . . . 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 282 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5 gcdcl 11562 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ0)
61, 5sylan2 282 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ0)
76nn0zd 9125 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8 simpl 108 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
91adantl 273 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10 nnne0 8708 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
1110neneqd 2304 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
1211intnand 899 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
1312adantl 273 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14 gcdn0cl 11558 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
158, 9, 13, 14syl21anc 1198 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
1615nnne0d 8725 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
17 dvdsval2 11403 . . . . . . . . . 10 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
187, 16, 8, 17syl3anc 1199 . . . . . . . . 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 984 . . . . . . 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 282 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
23 dvdsval2 11403 . . . . . . . . . . . 12 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
247, 16, 9, 23syl3anc 1199 . . . . . . . . . . 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 8687 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
2726adantl 273 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
286nn0red 8985 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
29 nngt0 8705 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
3029adantl 273 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
3115nngt0d 8724 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
3227, 28, 30, 31divgt0d 8653 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
3325, 32jca 302 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
34333adant3 984 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
35 elnnz 9018 . . . . . . . 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 4539 . . . . . . 7 (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
3820, 36, 37syl2anc 406 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
39 fveq2 5387 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1st𝑥) = (1st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩))
40 simp1 964 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑧 ∈ ℤ)
41153adant3 984 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 gcd 𝑛) ∈ ℕ)
42 znq 9368 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
4340, 41, 42syl2anc 406 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
4493adant3 984 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑛 ∈ ℤ)
45 znq 9368 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
4644, 41, 45syl2anc 406 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
47 op1stg 6014 . . . . . . . . . . 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 406 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (1st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑧 / (𝑧 gcd 𝑛)))
4939, 48sylan9eqr 2170 . . . . . . . . 9 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (1st𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
50 fveq2 5387 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2nd𝑥) = (2nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩))
51 op2ndg 6015 . . . . . . . . . . 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 406 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (2nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑛 / (𝑧 gcd 𝑛)))
5350, 52sylan9eqr 2170 . . . . . . . . 9 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (2nd𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
5449, 53oveq12d 5758 . . . . . . . 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 2124 . . . . . . 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 5758 . . . . . . . 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 2127 . . . . . . 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 462 . . . . . 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 11564 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ0)
6059nn0cnd 8986 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
61 1cnd 7746 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
626nn0cnd 8986 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
6315nnap0d 8726 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) # 0)
6462mulid1d 7747 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
65 zcn 9013 . . . . . . . . . . . . 13 (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
6665adantr 272 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
6766, 62, 63divcanap2d 8515 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
68 nncn 8688 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
6968adantl 273 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
7069, 62, 63divcanap2d 8515 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
7167, 70oveq12d 5758 . . . . . . . . . 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 11611 . . . . . . . . . . 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 1199 . . . . . . . . . 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 2155 . . . . . . . . 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 8387 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
76753adant3 984 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
77 nnap0 8709 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 # 0)
7877adantl 273 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 # 0)
7966, 69, 62, 78, 63divcanap7d 8542 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
8079eqeq2d 2127 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
8180biimp3ar 1307 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
8276, 81jca 302 . . . . . 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 2767 . . . . 5 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
84 elxp6 6033 . . . . . . 7 (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)))
85 elxp6 6033 . . . . . . 7 (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)))
86 simprl 503 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
8786ad2antrr 477 . . . . . . . . . . 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 504 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
8988ad2antrr 477 . . . . . . . . . . 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 509 . . . . . . . . . . 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 503 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (1st𝑦) ∈ ℤ)
9291ad2antlr 478 . . . . . . . . . . 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 504 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (2nd𝑦) ∈ ℕ)
9493ad2antlr 478 . . . . . . . . . . 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 511 . . . . . . . . . . 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 510 . . . . . . . . . . . 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 512 . . . . . . . . . . . 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 2150 . . . . . . . . . . 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 11684 . . . . . . . . . . 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 1224 . . . . . . . . . 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 2661 . . . . . . . . . . . 12 𝑥 ∈ V
102 1stexg 6031 . . . . . . . . . . . 12 (𝑥 ∈ V → (1st𝑥) ∈ V)
103101, 102ax-mp 5 . . . . . . . . . . 11 (1st𝑥) ∈ V
104 2ndexg 6032 . . . . . . . . . . . 12 (𝑥 ∈ V → (2nd𝑥) ∈ V)
105101, 104ax-mp 5 . . . . . . . . . . 11 (2nd𝑥) ∈ V
106103, 105opth 4127 . . . . . . . . . 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 505 . . . . . . . . 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 507 . . . . . . . . 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 2158 . . . . . . . 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 287 . . . . . 6 ((𝑥 ∈ (ℤ × ℕ) ∧ 𝑦 ∈ (ℤ × ℕ)) → (((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))
113112rgen2a 2461 . . . . 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 309 . . . 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 1166 . . 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 2530 . 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 9366 . 2 (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
118 fveq2 5387 . . . . . 6 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
119 fveq2 5387 . . . . . 6 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
120118, 119oveq12d 5758 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) gcd (2nd𝑥)) = ((1st𝑦) gcd (2nd𝑦)))
121120eqeq1d 2124 . . . 4 (𝑥 = 𝑦 → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((1st𝑦) gcd (2nd𝑦)) = 1))
122118, 119oveq12d 5758 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦)))
123122eqeq2d 2127 . . . 4 (𝑥 = 𝑦 → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑦) / (2nd𝑦))))
124121, 123anbi12d 462 . . 3 (𝑥 = 𝑦 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))))
125124reu4 2849 . 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 945   = wceq 1314  wcel 1463  wne 2283  wral 2391  wrex 2392  ∃!wreu 2393  Vcvv 2658  cop 3498   class class class wbr 3897   × cxp 4505  cfv 5091  (class class class)co 5740  1st c1st 6002  2nd c2nd 6003  cc 7582  cr 7583  0cc0 7584  1c1 7585   · cmul 7589   < clt 7764   # cap 8306   / cdiv 8395  cn 8680  0cn0 8931  cz 9008  cq 9363  cdvds 11400   gcd cgcd 11542
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-sup 6837  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-fl 9994  df-mod 10047  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-dvds 11401  df-gcd 11543
This theorem is referenced by:  qnumdencl  11771  qnumdenbi  11776
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