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Theorem qredeq 12079
Description: Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
Assertion
Ref Expression
qredeq (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 / 𝑁) = (𝑃 / 𝑄)) → (𝑀 = 𝑃𝑁 = 𝑄))

Proof of Theorem qredeq
StepHypRef Expression
1 zcn 9247 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
21adantr 276 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℂ)
3 nncn 8916 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
43adantl 277 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
5 nnap0 8937 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 # 0)
65adantl 277 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 # 0)
72, 4, 6divclapd 8736 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℂ)
873adant3 1017 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 / 𝑁) ∈ ℂ)
98adantr 276 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 / 𝑁) ∈ ℂ)
10 zcn 9247 . . . . . . . . . 10 (𝑃 ∈ ℤ → 𝑃 ∈ ℂ)
1110adantr 276 . . . . . . . . 9 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑃 ∈ ℂ)
12 nncn 8916 . . . . . . . . . 10 (𝑄 ∈ ℕ → 𝑄 ∈ ℂ)
1312adantl 277 . . . . . . . . 9 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 ∈ ℂ)
14 nnap0 8937 . . . . . . . . . 10 (𝑄 ∈ ℕ → 𝑄 # 0)
1514adantl 277 . . . . . . . . 9 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 # 0)
1611, 13, 15divclapd 8736 . . . . . . . 8 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℂ)
17163adant3 1017 . . . . . . 7 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 / 𝑄) ∈ ℂ)
1817adantl 277 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑃 / 𝑄) ∈ ℂ)
1933ad2ant2 1019 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℂ)
2019adantr 276 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℂ)
2153ad2ant2 1019 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 # 0)
2221adantr 276 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 # 0)
239, 18, 20, 22mulcanapd 8607 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑀 / 𝑁)) = (𝑁 · (𝑃 / 𝑄)) ↔ (𝑀 / 𝑁) = (𝑃 / 𝑄)))
242, 4, 6divcanap2d 8738 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
25243adant3 1017 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
2625adantr 276 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
2726eqeq1d 2186 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑀 / 𝑁)) = (𝑁 · (𝑃 / 𝑄)) ↔ 𝑀 = (𝑁 · (𝑃 / 𝑄))))
2823, 27bitr3d 190 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 / 𝑁) = (𝑃 / 𝑄) ↔ 𝑀 = (𝑁 · (𝑃 / 𝑄))))
2913ad2ant1 1018 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℂ)
3029adantr 276 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℂ)
31 mulcl 7929 . . . . . . 7 ((𝑁 ∈ ℂ ∧ (𝑃 / 𝑄) ∈ ℂ) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
3219, 17, 31syl2an 289 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
33123ad2ant2 1019 . . . . . . 7 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℂ)
3433adantl 277 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℂ)
35143ad2ant2 1019 . . . . . . 7 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 # 0)
3635adantl 277 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 # 0)
3730, 32, 34, 36mulcanap2d 8608 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑄) = ((𝑁 · (𝑃 / 𝑄)) · 𝑄) ↔ 𝑀 = (𝑁 · (𝑃 / 𝑄))))
3820, 18, 34mulassd 7971 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑃 / 𝑄)) · 𝑄) = (𝑁 · ((𝑃 / 𝑄) · 𝑄)))
3911, 13, 15divcanap1d 8737 . . . . . . . . . 10 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
40393adant3 1017 . . . . . . . . 9 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
4140adantl 277 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
4241oveq2d 5885 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · ((𝑃 / 𝑄) · 𝑄)) = (𝑁 · 𝑃))
4338, 42eqtrd 2210 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑃 / 𝑄)) · 𝑄) = (𝑁 · 𝑃))
4443eqeq2d 2189 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑄) = ((𝑁 · (𝑃 / 𝑄)) · 𝑄) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
4537, 44bitr3d 190 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 = (𝑁 · (𝑃 / 𝑄)) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
4628, 45bitrd 188 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 / 𝑁) = (𝑃 / 𝑄) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
47 nnz 9261 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48473ad2ant2 1019 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℤ)
49 simp2 998 . . . . . . . . 9 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℕ)
5048, 49anim12i 338 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ))
5150adantr 276 . . . . . . 7 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ))
5248adantr 276 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℤ)
53 simpl1 1000 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℤ)
54 nnz 9261 . . . . . . . . . . . 12 (𝑄 ∈ ℕ → 𝑄 ∈ ℤ)
55543ad2ant2 1019 . . . . . . . . . . 11 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℤ)
5655adantl 277 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℤ)
5752, 53, 563jca 1177 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ))
5857adantr 276 . . . . . . . 8 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ))
59 simp1 997 . . . . . . . . . . . 12 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℤ)
60 dvdsmul1 11804 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑃))
6148, 59, 60syl2an 289 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∥ (𝑁 · 𝑃))
6261adantr 276 . . . . . . . . . 10 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∥ (𝑁 · 𝑃))
63 simpr 110 . . . . . . . . . 10 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑀 · 𝑄) = (𝑁 · 𝑃))
6462, 63breqtrrd 4028 . . . . . . . . 9 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∥ (𝑀 · 𝑄))
65 gcdcom 11957 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
6647, 65sylan 283 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
6766ancoms 268 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
68673adant3 1017 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
69 simp3 999 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 gcd 𝑁) = 1)
7068, 69eqtrd 2210 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = 1)
7170ad2antrr 488 . . . . . . . . 9 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 gcd 𝑀) = 1)
7264, 71jca 306 . . . . . . . 8 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 ∥ (𝑀 · 𝑄) ∧ (𝑁 gcd 𝑀) = 1))
73 coprmdvds 12075 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → ((𝑁 ∥ (𝑀 · 𝑄) ∧ (𝑁 gcd 𝑀) = 1) → 𝑁𝑄))
7458, 72, 73sylc 62 . . . . . . 7 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁𝑄)
75 dvdsle 11833 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑁𝑄𝑁𝑄))
7651, 74, 75sylc 62 . . . . . 6 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁𝑄)
77 simp2 998 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℕ)
7855, 77anim12i 338 . . . . . . . . 9 (((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
7978ancoms 268 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
8079adantr 276 . . . . . . 7 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
81 simpr1 1003 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℤ)
8256, 81, 523jca 1177 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
8382adantr 276 . . . . . . . 8 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
84 simp1 997 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℤ)
85 dvdsmul2 11805 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑀 · 𝑄))
8684, 55, 85syl2an 289 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∥ (𝑀 · 𝑄))
8786adantr 276 . . . . . . . . . 10 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∥ (𝑀 · 𝑄))
88103ad2ant1 1018 . . . . . . . . . . . . 13 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℂ)
89 mulcom 7931 . . . . . . . . . . . . 13 ((𝑁 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
9019, 88, 89syl2an 289 . . . . . . . . . . . 12 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
9190adantr 276 . . . . . . . . . . 11 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
9263, 91eqtrd 2210 . . . . . . . . . 10 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑀 · 𝑄) = (𝑃 · 𝑁))
9387, 92breqtrd 4026 . . . . . . . . 9 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∥ (𝑃 · 𝑁))
94 gcdcom 11957 . . . . . . . . . . . . . 14 ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
9554, 94sylan 283 . . . . . . . . . . . . 13 ((𝑄 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
9695ancoms 268 . . . . . . . . . . . 12 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
97963adant3 1017 . . . . . . . . . . 11 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
98 simp3 999 . . . . . . . . . . 11 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 gcd 𝑄) = 1)
9997, 98eqtrd 2210 . . . . . . . . . 10 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = 1)
10099ad2antlr 489 . . . . . . . . 9 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 gcd 𝑃) = 1)
10193, 100jca 306 . . . . . . . 8 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 ∥ (𝑃 · 𝑁) ∧ (𝑄 gcd 𝑃) = 1))
102 coprmdvds 12075 . . . . . . . 8 ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑁) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄𝑁))
10383, 101, 102sylc 62 . . . . . . 7 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄𝑁)
104 dvdsle 11833 . . . . . . 7 ((𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑄𝑁𝑄𝑁))
10580, 103, 104sylc 62 . . . . . 6 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄𝑁)
106 nnre 8915 . . . . . . . . 9 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
1071063ad2ant2 1019 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℝ)
108107ad2antrr 488 . . . . . . 7 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∈ ℝ)
109 nnre 8915 . . . . . . . . 9 (𝑄 ∈ ℕ → 𝑄 ∈ ℝ)
1101093ad2ant2 1019 . . . . . . . 8 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℝ)
111110ad2antlr 489 . . . . . . 7 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∈ ℝ)
112108, 111letri3d 8063 . . . . . 6 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 = 𝑄 ↔ (𝑁𝑄𝑄𝑁)))
11376, 105, 112mpbir2and 944 . . . . 5 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 = 𝑄)
114 oveq2 5877 . . . . . . . . . 10 (𝑁 = 𝑄 → (𝑀 · 𝑁) = (𝑀 · 𝑄))
115114eqeq1d 2186 . . . . . . . . 9 (𝑁 = 𝑄 → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
116115anbi2d 464 . . . . . . . 8 (𝑁 = 𝑄 → ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑁) = (𝑁 · 𝑃)) ↔ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃))))
117 mulcom 7931 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
1181, 3, 117syl2an 289 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
1191183adant3 1017 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
120119adantr 276 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
121120eqeq1d 2186 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ (𝑁 · 𝑀) = (𝑁 · 𝑃)))
12288adantl 277 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℂ)
12330, 122, 20, 22mulcanapd 8607 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · 𝑀) = (𝑁 · 𝑃) ↔ 𝑀 = 𝑃))
124121, 123bitrd 188 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ 𝑀 = 𝑃))
125124biimpa 296 . . . . . . . 8 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑁) = (𝑁 · 𝑃)) → 𝑀 = 𝑃)
126116, 125syl6bir 164 . . . . . . 7 (𝑁 = 𝑄 → ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑀 = 𝑃))
127126com12 30 . . . . . 6 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 = 𝑄𝑀 = 𝑃))
128127ancrd 326 . . . . 5 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 = 𝑄 → (𝑀 = 𝑃𝑁 = 𝑄)))
129113, 128mpd 13 . . . 4 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑀 = 𝑃𝑁 = 𝑄))
130129ex 115 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑄) = (𝑁 · 𝑃) → (𝑀 = 𝑃𝑁 = 𝑄)))
13146, 130sylbid 150 . 2 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 / 𝑁) = (𝑃 / 𝑄) → (𝑀 = 𝑃𝑁 = 𝑄)))
1321313impia 1200 1 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 / 𝑁) = (𝑃 / 𝑄)) → (𝑀 = 𝑃𝑁 = 𝑄))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148   class class class wbr 4000  (class class class)co 5869  cc 7800  cr 7801  0cc0 7802  1c1 7803   · cmul 7807  cle 7983   # cap 8528   / cdiv 8618  cn 8908  cz 9242  cdvds 11778   gcd cgcd 11926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927
This theorem is referenced by:  qredeu  12080
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