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Theorem rplpwr 11998
Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rplpwr ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1))

Proof of Theorem rplpwr
Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5876 . . . . . . . 8 (𝑘 = 1 → (𝐴𝑘) = (𝐴↑1))
21oveq1d 5883 . . . . . . 7 (𝑘 = 1 → ((𝐴𝑘) gcd 𝐵) = ((𝐴↑1) gcd 𝐵))
32eqeq1d 2186 . . . . . 6 (𝑘 = 1 → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑1) gcd 𝐵) = 1))
43imbi2d 230 . . . . 5 (𝑘 = 1 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)))
5 oveq2 5876 . . . . . . . 8 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
65oveq1d 5883 . . . . . . 7 (𝑘 = 𝑛 → ((𝐴𝑘) gcd 𝐵) = ((𝐴𝑛) gcd 𝐵))
76eqeq1d 2186 . . . . . 6 (𝑘 = 𝑛 → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴𝑛) gcd 𝐵) = 1))
87imbi2d 230 . . . . 5 (𝑘 = 𝑛 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) gcd 𝐵) = 1)))
9 oveq2 5876 . . . . . . . 8 (𝑘 = (𝑛 + 1) → (𝐴𝑘) = (𝐴↑(𝑛 + 1)))
109oveq1d 5883 . . . . . . 7 (𝑘 = (𝑛 + 1) → ((𝐴𝑘) gcd 𝐵) = ((𝐴↑(𝑛 + 1)) gcd 𝐵))
1110eqeq1d 2186 . . . . . 6 (𝑘 = (𝑛 + 1) → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
1211imbi2d 230 . . . . 5 (𝑘 = (𝑛 + 1) → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
13 oveq2 5876 . . . . . . . 8 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
1413oveq1d 5883 . . . . . . 7 (𝑘 = 𝑁 → ((𝐴𝑘) gcd 𝐵) = ((𝐴𝑁) gcd 𝐵))
1514eqeq1d 2186 . . . . . 6 (𝑘 = 𝑁 → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴𝑁) gcd 𝐵) = 1))
1615imbi2d 230 . . . . 5 (𝑘 = 𝑁 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd 𝐵) = 1)))
17 nncn 8903 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
1817exp1d 10621 . . . . . . . . 9 (𝐴 ∈ ℕ → (𝐴↑1) = 𝐴)
1918oveq1d 5883 . . . . . . . 8 (𝐴 ∈ ℕ → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
2019adantr 276 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
2120eqeq1d 2186 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝐴↑1) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
2221biimpar 297 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)
23 df-3an 980 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ))
24 simpl1 1000 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
2524nncnd 8909 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℂ)
26 simpl3 1002 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ)
2726nnnn0d 9205 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ0)
2825, 27expp1d 10627 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = ((𝐴𝑛) · 𝐴))
29 simp1 997 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℕ)
30 nnnn0 9159 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
31303ad2ant3 1020 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
3229, 31nnexpcld 10648 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℕ)
3332nnzd 9350 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℤ)
3433adantr 276 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑛) ∈ ℤ)
3534zcnd 9352 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑛) ∈ ℂ)
3635, 25mulcomd 7956 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) · 𝐴) = (𝐴 · (𝐴𝑛)))
3728, 36eqtrd 2210 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = (𝐴 · (𝐴𝑛)))
3837oveq2d 5884 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴 · (𝐴𝑛))))
39 simpl2 1001 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
4032adantr 276 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑛) ∈ ℕ)
41 nnz 9248 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
42413ad2ant1 1018 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
43 nnz 9248 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
44433ad2ant2 1019 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℤ)
45 gcdcom 11944 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
4642, 44, 45syl2anc 411 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
4746eqeq1d 2186 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐴) = 1))
4847biimpa 296 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd 𝐴) = 1)
49 rpmulgcd 11997 . . . . . . . . . . . . . 14 (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) ∧ (𝐵 gcd 𝐴) = 1) → (𝐵 gcd (𝐴 · (𝐴𝑛))) = (𝐵 gcd (𝐴𝑛)))
5039, 24, 40, 48, 49syl31anc 1241 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴 · (𝐴𝑛))) = (𝐵 gcd (𝐴𝑛)))
5138, 50eqtrd 2210 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴𝑛)))
52 peano2nn 8907 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
53523ad2ant3 1020 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
5453adantr 276 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ)
5554nnnn0d 9205 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ0)
5624, 55nnexpcld 10648 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℕ)
5756nnzd 9350 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℤ)
5844adantr 276 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℤ)
59 gcdcom 11944 . . . . . . . . . . . . 13 (((𝐴↑(𝑛 + 1)) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1))))
6057, 58, 59syl2anc 411 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1))))
61 gcdcom 11944 . . . . . . . . . . . . 13 (((𝐴𝑛) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴𝑛) gcd 𝐵) = (𝐵 gcd (𝐴𝑛)))
6234, 58, 61syl2anc 411 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) gcd 𝐵) = (𝐵 gcd (𝐴𝑛)))
6351, 60, 623eqtr4d 2220 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = ((𝐴𝑛) gcd 𝐵))
6463eqeq1d 2186 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1 ↔ ((𝐴𝑛) gcd 𝐵) = 1))
6564biimprd 158 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
6623, 65sylanbr 285 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
6766an32s 568 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑛 ∈ ℕ) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
6867expcom 116 . . . . . 6 (𝑛 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
6968a2d 26 . . . . 5 (𝑛 ∈ ℕ → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) gcd 𝐵) = 1) → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
704, 8, 12, 16, 22, 69nnind 8911 . . . 4 (𝑁 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd 𝐵) = 1))
7170expd 258 . . 3 (𝑁 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1)))
7271com12 30 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1)))
73723impia 1200 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  (class class class)co 5868  1c1 7790   + caddc 7792   · cmul 7794  cn 8895  0cn0 9152  cz 9229  cexp 10492   gcd cgcd 11913
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 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
This theorem depends on definitions:  df-bi 117  df-stab 831  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 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-sup 6976  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-fz 9983  df-fzo 10116  df-fl 10243  df-mod 10296  df-seqfrec 10419  df-exp 10493  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979  df-dvds 11766  df-gcd 11914
This theorem is referenced by:  rppwr  11999  logbgcd1irr  14018  logbgcd1irraplemexp  14019  lgsne0  14072  2sqlem8  14092
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