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Theorem zq 8657
 Description: An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
Assertion
Ref Expression
zq (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)

Proof of Theorem zq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zcn 8306 . . . . . . 7 (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
21div1d 7830 . . . . . 6 (𝑥 ∈ ℤ → (𝑥 / 1) = 𝑥)
32eqeq2d 2067 . . . . 5 (𝑥 ∈ ℤ → (𝐴 = (𝑥 / 1) ↔ 𝐴 = 𝑥))
4 eqcom 2058 . . . . 5 (𝑥 = 𝐴𝐴 = 𝑥)
53, 4syl6rbbr 192 . . . 4 (𝑥 ∈ ℤ → (𝑥 = 𝐴𝐴 = (𝑥 / 1)))
6 1nn 8000 . . . . 5 1 ∈ ℕ
7 oveq2 5547 . . . . . . 7 (𝑦 = 1 → (𝑥 / 𝑦) = (𝑥 / 1))
87eqeq2d 2067 . . . . . 6 (𝑦 = 1 → (𝐴 = (𝑥 / 𝑦) ↔ 𝐴 = (𝑥 / 1)))
98rspcev 2673 . . . . 5 ((1 ∈ ℕ ∧ 𝐴 = (𝑥 / 1)) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
106, 9mpan 408 . . . 4 (𝐴 = (𝑥 / 1) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
115, 10syl6bi 156 . . 3 (𝑥 ∈ ℤ → (𝑥 = 𝐴 → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)))
1211reximia 2431 . 2 (∃𝑥 ∈ ℤ 𝑥 = 𝐴 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
13 risset 2369 . 2 (𝐴 ∈ ℤ ↔ ∃𝑥 ∈ ℤ 𝑥 = 𝐴)
14 elq 8653 . 2 (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
1512, 13, 143imtr4i 194 1 (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)