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Theorem qredeu 16692
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 12632 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2 gcddvds 16537 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
32simpld 494 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
41, 3sylan2 593 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5 gcdcl 16540 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ0)
61, 5sylan2 593 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ0)
76nn0zd 12637 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8 simpl 482 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
91adantl 481 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10 nnne0 12298 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
1110neneqd 2943 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
1211intnand 488 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
1312adantl 481 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14 gcdn0cl 16536 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
158, 9, 13, 14syl21anc 838 . . . . . . . . . . 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 16290 . . . . . . . . . 10 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
197, 17, 8, 18syl3anc 1370 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
204, 19mpbid 232 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21203adant3 1131 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
222simprd 495 . . . . . . . . . . . 12 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
231, 22sylan2 593 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
24 dvdsval2 16290 . . . . . . . . . . . 12 (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
257, 17, 9, 24syl3anc 1370 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
2623, 25mpbid 232 . . . . . . . . . 10 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
27 nnre 12271 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
2827adantl 481 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
296nn0red 12586 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
30 nngt0 12295 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
3130adantl 481 . . . . . . . . . . 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 511 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36353adant3 1131 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
37 elnnz 12621 . . . . . . . 8 ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
3836, 37sylibr 234 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
3921, 38opelxpd 5728 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
4020, 26gcdcld 16542 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ0)
4140nn0cnd 12587 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
42 1cnd 11254 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
436nn0cnd 12587 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
4443mulridd 11276 . . . . . . . . 9 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
45 zcn 12616 . . . . . . . . . . . 12 (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
4645adantr 480 . . . . . . . . . . 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 481 . . . . . . . . . . 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 7449 . . . . . . . . 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 16582 . . . . . . . . . 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 1370 . . . . . . . . 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 2782 . . . . . . . 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 11896 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
56553adant3 1131 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
5710adantl 481 . . . . . . . . 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 2746 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
6059biimp3ar 1469 . . . . . 6 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
61 ovex 7464 . . . . . . . . . . 11 (𝑧 / (𝑧 gcd 𝑛)) ∈ V
62 ovex 7464 . . . . . . . . . . 11 (𝑛 / (𝑧 gcd 𝑛)) ∈ V
6361, 62op1std 8023 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1st𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
6461, 62op2ndd 8024 . . . . . . . . . 10 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2nd𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
6563, 64oveq12d 7449 . . . . . . . . 9 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1st𝑥) gcd (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))
6665eqeq1d 2737 . . . . . . . 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 7449 . . . . . . . . 9 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1st𝑥) / (2nd𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
6867eqeq2d 2746 . . . . . . . 8 (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
6966, 68anbi12d 632 . . . . . . 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 3622 . . . . . 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 837 . . . . 5 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))
72 elxp6 8047 . . . . . . 7 (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)))
73 elxp6 8047 . . . . . . 7 (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)))
74 simprl 771 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
7574ad2antrr 726 . . . . . . . . . . 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 773 . . . . . . . . . . . 12 ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℤ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
7776ad2antrr 726 . . . . . . . . . . 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 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))))) → ((1st𝑥) gcd (2nd𝑥)) = 1)
79 simprl 771 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (1st𝑦) ∈ ℤ)
8079ad2antlr 727 . . . . . . . . . . 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 773 . . . . . . . . . . . 12 ((𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ ℤ ∧ (2nd𝑦) ∈ ℕ)) → (2nd𝑦) ∈ ℕ)
8281ad2antlr 727 . . . . . . . . . . 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 781 . . . . . . . . . . 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 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 = ((1st𝑥) / (2nd𝑥)))
85 simprrr 782 . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 16691 . . . . . . . . . . 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 1394 . . . . . . . . . 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 6920 . . . . . . . . . . 11 (1st𝑥) ∈ V
90 fvex 6920 . . . . . . . . . . 11 (2nd𝑥) ∈ V
9189, 90opth 5487 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
9288, 91sylibr 234 . . . . . . . . 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 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))))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
94 simplrl 777 . . . . . . . . 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 2785 . . . . . . . 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 412 . . . . . . 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 598 . . . . . 6 ((𝑥 ∈ (ℤ × ℕ) ∧ 𝑦 ∈ (ℤ × ℕ)) → (((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ∧ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))) → 𝑥 = 𝑦))
9897rgen2 3197 . . . . 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 520 . . . 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 1120 . . 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 3199 . 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 12990 . 2 (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
103 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
104 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
105103, 104oveq12d 7449 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) gcd (2nd𝑥)) = ((1st𝑦) gcd (2nd𝑦)))
106105eqeq1d 2737 . . . 4 (𝑥 = 𝑦 → (((1st𝑥) gcd (2nd𝑥)) = 1 ↔ ((1st𝑦) gcd (2nd𝑦)) = 1))
107103, 104oveq12d 7449 . . . . 5 (𝑥 = 𝑦 → ((1st𝑥) / (2nd𝑥)) = ((1st𝑦) / (2nd𝑦)))
108107eqeq2d 2746 . . . 4 (𝑥 = 𝑦 → (𝐴 = ((1st𝑥) / (2nd𝑥)) ↔ 𝐴 = ((1st𝑦) / (2nd𝑦))))
109106, 108anbi12d 632 . . 3 (𝑥 = 𝑦 → ((((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))) ↔ (((1st𝑦) gcd (2nd𝑦)) = 1 ∧ 𝐴 = ((1st𝑦) / (2nd𝑦)))))
110109reu4 3740 . 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 292 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  ∃!wreu 3376  cop 4637   class class class wbr 5148   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  cc 11151  cr 11152  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293   / cdiv 11918  cn 12264  0cn0 12524  cz 12611  cq 12988  cdvds 16287   gcd cgcd 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529
This theorem is referenced by:  qnumdencl  16773  qnumdenbi  16778
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