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Theorem qredeu 15579
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 11601 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2 gcddvds 15433 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
32simpld 482 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
41, 3sylan2 580 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5 gcdcl 15436 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ0)
61, 5sylan2 580 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ0)
76nn0zd 11682 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8 simpl 468 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
91adantl 467 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10 nnne0 11255 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
1110neneqd 2948 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
1211intnand 476 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
1312adantl 467 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14 gcdn0cl 15432 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
158, 9, 13, 14syl21anc 1475 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
16 nnne0 11255 . . . . . . . . . . 11 ((𝑧 gcd 𝑛) ∈ ℕ → (𝑧 gcd 𝑛) ≠ 0)
1715, 16syl 17 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
18 dvdsval2 15192 . . . . . . . . . 10 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
197, 17, 8, 18syl3anc 1476 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
204, 19mpbid 222 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21203adant3 1126 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
222simprd 483 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
231, 22sylan2 580 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
24 dvdsval2 15192 . . . . . . . . . . . 12 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
257, 17, 9, 24syl3anc 1476 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
2623, 25mpbid 222 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
27 nnre 11229 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
2827adantl 467 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
296nn0red 11554 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
30 nngt0 11251 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
3130adantl 467 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
32 nngt0 11251 . . . . . . . . . . . 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 11161 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
3526, 34jca 501 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36353adant3 1126 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
37 elnnz 11589 . . . . . . . 8 ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
3836, 37sylibr 224 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
3921, 38opelxpd 5289 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
4020, 26gcdcld 15438 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ0)
4140nn0cnd 11555 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
42 1cnd 10258 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
436nn0cnd 11555 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
4443mulid1d 10259 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
45 zcn 11584 . . . . . . . . . . . 12 (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
4645adantr 466 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
4746, 43, 17divcan2d 11005 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
48 nncn 11230 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
4948adantl 467 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
5049, 43, 17divcan2d 11005 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
5147, 50oveq12d 6811 . . . . . . . . 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 15473 . . . . . . . . . 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 1476 . . . . . . . . 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 2812 . . . . . . . 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 10864 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
56553adant3 1126 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
5710adantl 467 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
5846, 49, 43, 57, 17divcan7d 11031 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
5958eqeq2d 2781 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
6059biimp3ar 1581 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
61 ovex 6823 . . . . . . . . . . 11 (𝑧 / (𝑧 gcd 𝑛)) ∈ V
62 ovex 6823 . . . . . . . . . . 11 (𝑛 / (𝑧 gcd 𝑛)) ∈ V
6361, 62op1std 7325 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1st𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
6461, 62op2ndd 7326 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2nd𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
6563, 64oveq12d 6811 . . . . . . . . 9 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1st𝑥) gcd (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))
6665eqeq1d 2773 . . . . . . . 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 6811 . . . . . . . . 9 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1st𝑥) / (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
6867eqeq2d 2781 . . . . . . . 8 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
6966, 68anbi12d 616 . . . . . . 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 3460 . . . . . 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 1474 . . . . 5 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
72 elxp6 7349 . . . . . . 7 (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)))
73 elxp6 7349 . . . . . . 7 (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)))
74 simprl 754 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
7574ad2antrr 705 . . . . . . . . . . 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 756 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
7776ad2antrr 705 . . . . . . . . . . 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 764 . . . . . . . . . . 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 754 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (1st𝑦) ∈ ℤ)
8079ad2antlr 706 . . . . . . . . . . 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 756 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (2nd𝑦) ∈ ℕ)
8281ad2antlr 706 . . . . . . . . . . 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 766 . . . . . . . . . . 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 765 . . . . . . . . . . . 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 767 . . . . . . . . . . . 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 2807 . . . . . . . . . . 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 15578 . . . . . . . . . . 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 1501 . . . . . . . . . 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 6342 . . . . . . . . . . 11 (1st𝑥) ∈ V
90 fvex 6342 . . . . . . . . . . 11 (2nd𝑥) ∈ V
9189, 90opth 5072 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
9288, 91sylibr 224 . . . . . . . . 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 758 . . . . . . . . 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 762 . . . . . . . . 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 2815 . . . . . . . 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 397 . . . . . . 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 585 . . . . . 6 ((𝑥 ∈ (ℤ × ℕ) ∧ 𝑦 ∈ (ℤ × ℕ)) → (((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))
9897rgen2a 3126 . . . . 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 510 . . . 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 1114 . . 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 3184 . 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 11993 . 2 (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
103 fveq2 6332 . . . . . 6 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
104 fveq2 6332 . . . . . 6 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
105103, 104oveq12d 6811 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) gcd (2nd𝑥)) = ((1st𝑦) gcd (2nd𝑦)))
106105eqeq1d 2773 . . . 4 (𝑥 = 𝑦 → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((1st𝑦) gcd (2nd𝑦)) = 1))
107103, 104oveq12d 6811 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦)))
108107eqeq2d 2781 . . . 4 (𝑥 = 𝑦 → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑦) / (2nd𝑦))))
109106, 108anbi12d 616 . . 3 (𝑥 = 𝑦 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))))
110109reu4 3552 . 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 281 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  ∃!wreu 3063  cop 4322   class class class wbr 4786   × cxp 5247  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  cc 10136  cr 10137  0cc0 10138  1c1 10139   · cmul 10143   < clt 10276   / cdiv 10886  cn 11222  0cn0 11494  cz 11579  cq 11991  cdvds 15189   gcd cgcd 15424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-gcd 15425
This theorem is referenced by:  qnumdencl  15654  qnumdenbi  15659
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