Theorem znq 4630

Description: The ratio of an integer and a natural number is a rational number.

Assertion

Ref	Expression
znq	⊢ ((A ∈ ℤ ∧ B ∈ ℕ) → (A / B) ∈ ℚ)

Proof of Theorem znq

Step	Hyp	Ref	Expression
1		cleqid 1102	. . 3 ⊢ (A / B) = (A / B)
2		opreq1 3006	. . . . 5 ⊢ (x = A → (x / y) = (A / y))
3	2	cleq2d 1112	. . . 4 ⊢ (x = A → ((A / B) = (x / y) ↔ (A / B) = (A / y)))
4		opreq2 3007	. . . . 5 ⊢ (y = B → (A / y) = (A / B))
5	4	cleq2d 1112	. . . 4 ⊢ (y = B → ((A / B) = (A / y) ↔ (A / B) = (A / B)))
6	3, 5	rcla42ev 1405	. . 3 ⊢ (((A ∈ ℤ ∧ B ∈ ℕ) ∧ (A / B) = (A / B)) → ∃x ∈ ℤ ∃y ∈ ℕ (A / B) = (x / y))
7	1, 6	mpan2 519	. 2 ⊢ ((A ∈ ℤ ∧ B ∈ ℕ) → ∃x ∈ ℤ ∃y ∈ ℕ (A / B) = (x / y))
8		elq 4629	. 2 ⊢ ((A / B) ∈ ℚ ↔ ∃x ∈ ℤ ∃y ∈ ℕ (A / B) = (x / y))
9	7, 8	sylibr 175	1 ⊢ ((A ∈ ℤ ∧ B ∈ ℕ) → (A / B) ∈ ℚ)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  (class class class)co 3001   / cdiv 4091  ℕcn 4093  ℤcz 4095  ℚcq 4096

This theorem is referenced by:  nnrecqt 4649  qbtwnre 4650  nthruc 4784

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-q 4628

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