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Theorem qredeu 16659
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 12631 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2 gcddvds 16503 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
32simpld 493 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
41, 3sylan2 591 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5 gcdcl 16506 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ0)
61, 5sylan2 591 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ0)
76nn0zd 12636 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8 simpl 481 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
91adantl 480 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10 nnne0 12298 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
1110neneqd 2935 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
1211intnand 487 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
1312adantl 480 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14 gcdn0cl 16502 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
158, 9, 13, 14syl21anc 836 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
16 nnne0 12298 . . . . . . . . . . 11 ((𝑧 gcd 𝑛) ∈ ℕ → (𝑧 gcd 𝑛) ≠ 0)
1715, 16syl 17 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
18 dvdsval2 16259 . . . . . . . . . 10 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
197, 17, 8, 18syl3anc 1368 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
204, 19mpbid 231 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21203adant3 1129 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
222simprd 494 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
231, 22sylan2 591 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
24 dvdsval2 16259 . . . . . . . . . . . 12 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
257, 17, 9, 24syl3anc 1368 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
2623, 25mpbid 231 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
27 nnre 12271 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
2827adantl 480 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
296nn0red 12585 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
30 nngt0 12295 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
3130adantl 480 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
32 nngt0 12295 . . . . . . . . . . . 12 ((𝑧 gcd 𝑛) ∈ ℕ → 0 < (𝑧 gcd 𝑛))
3315, 32syl 17 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
3428, 29, 31, 33divgt0d 12201 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
3526, 34jca 510 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36353adant3 1129 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
37 elnnz 12620 . . . . . . . 8 ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
3836, 37sylibr 233 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
3921, 38opelxpd 5721 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
4020, 26gcdcld 16508 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ0)
4140nn0cnd 12586 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
42 1cnd 11259 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
436nn0cnd 12586 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
4443mulridd 11281 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
45 zcn 12615 . . . . . . . . . . . 12 (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
4645adantr 479 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
4746, 43, 17divcan2d 12043 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
48 nncn 12272 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
4948adantl 480 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
5049, 43, 17divcan2d 12043 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
5147, 50oveq12d 7442 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = (𝑧 gcd 𝑛))
52 mulgcd 16549 . . . . . . . . . 10 (((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)))))
536, 20, 26, 52syl3anc 1368 . . . . . . . . 9 ((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)))))
5444, 51, 533eqtr2rd 2773 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · 1))
5541, 42, 43, 17, 54mulcanad 11899 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
56553adant3 1129 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
5710adantl 480 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
5846, 49, 43, 57, 17divcan7d 12069 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
5958eqeq2d 2737 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
6059biimp3ar 1467 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
61 ovex 7457 . . . . . . . . . . 11 (𝑧 / (𝑧 gcd 𝑛)) ∈ V
62 ovex 7457 . . . . . . . . . . 11 (𝑛 / (𝑧 gcd 𝑛)) ∈ V
6361, 62op1std 8013 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1st𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
6461, 62op2ndd 8014 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2nd𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
6563, 64oveq12d 7442 . . . . . . . . 9 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1st𝑥) gcd (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))
6665eqeq1d 2728 . . . . . . . 8 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1))
6763, 64oveq12d 7442 . . . . . . . . 9 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1st𝑥) / (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
6867eqeq2d 2737 . . . . . . . 8 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
6966, 68anbi12d 630 . . . . . . 7 (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))))))
7069rspcev 3608 . . . . . 6 ((⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ) ∧ (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))) → ∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
7139, 56, 60, 70syl12anc 835 . . . . 5 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
72 elxp6 8037 . . . . . . 7 (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)))
73 elxp6 8037 . . . . . . 7 (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)))
74 simprl 769 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
7574ad2antrr 724 . . . . . . . . . . 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)
76 simprr 771 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
7776ad2antrr 724 . . . . . . . . . . 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)
78 simprll 777 . . . . . . . . . . 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)
79 simprl 769 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (1st𝑦) ∈ ℤ)
8079ad2antlr 725 . . . . . . . . . . 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)
81 simprr 771 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (2nd𝑦) ∈ ℕ)
8281ad2antlr 725 . . . . . . . . . . 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)
83 simprrl 779 . . . . . . . . . . 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)
84 simprlr 778 . . . . . . . . . . . 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)))
85 simprrr 780 . . . . . . . . . . . 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)))
8684, 85eqtr3d 2768 . . . . . . . . . . 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)))
87 qredeq 16658 . . . . . . . . . . 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)))
8875, 77, 78, 80, 82, 83, 86, 87syl331anc 1392 . . . . . . . . . 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)))
89 fvex 6914 . . . . . . . . . . 11 (1st𝑥) ∈ V
90 fvex 6914 . . . . . . . . . . 11 (2nd𝑥) ∈ V
9189, 90opth 5482 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
9288, 91sylibr 233 . . . . . . . . 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)⟩)
93 simplll 773 . . . . . . . . 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)⟩)
94 simplrl 775 . . . . . . . . 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)⟩)
9592, 93, 943eqtr4d 2776 . . . . . . . 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)
9695ex 411 . . . . . . 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))
9772, 73, 96syl2anb 596 . . . . . 6 ((𝑥 ∈ (ℤ × ℕ) ∧ 𝑦 ∈ (ℤ × ℕ)) → (((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))
9897rgen2 3188 . . . . 5 𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦)
9971, 98jctir 519 . . . 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)))
100993expia 1118 . . 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))))
101100rexlimivv 3190 . 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)))
102 elq 12986 . 2 (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
103 fveq2 6901 . . . . . 6 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
104 fveq2 6901 . . . . . 6 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
105103, 104oveq12d 7442 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) gcd (2nd𝑥)) = ((1st𝑦) gcd (2nd𝑦)))
106105eqeq1d 2728 . . . 4 (𝑥 = 𝑦 → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((1st𝑦) gcd (2nd𝑦)) = 1))
107103, 104oveq12d 7442 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦)))
108107eqeq2d 2737 . . . 4 (𝑥 = 𝑦 → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑦) / (2nd𝑦))))
109106, 108anbi12d 630 . . 3 (𝑥 = 𝑦 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))))
110109reu4 3725 . 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)))
111101, 102, 1103imtr4i 291 1 (𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  ∃!wreu 3362  cop 4639   class class class wbr 5153   × cxp 5680  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  cc 11156  cr 11157  0cc0 11158  1c1 11159   · cmul 11163   < clt 11298   / cdiv 11921  cn 12264  0cn0 12524  cz 12610  cq 12984  cdvds 16256   gcd cgcd 16494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-fl 13812  df-mod 13890  df-seq 14022  df-exp 14082  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-dvds 16257  df-gcd 16495
This theorem is referenced by:  qnumdencl  16741  qnumdenbi  16746
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