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Theorem fvprmselgcd1 17008
Description: The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.)
Hypothesis
Ref Expression
fvprmselelfz.f 𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
Assertion
Ref Expression
fvprmselgcd1 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
Distinct variable groups:   𝑚,𝑁   𝑚,𝑋   𝑚,𝑌
Allowed substitution hint:   𝐹(𝑚)

Proof of Theorem fvprmselgcd1
StepHypRef Expression
1 fvprmselelfz.f . . . . . 6 𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
2 eleq1 2813 . . . . . . . 8 (𝑚 = 𝑋 → (𝑚 ∈ ℙ ↔ 𝑋 ∈ ℙ))
3 id 22 . . . . . . . 8 (𝑚 = 𝑋𝑚 = 𝑋)
42, 3ifbieq1d 4549 . . . . . . 7 (𝑚 = 𝑋 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑋 ∈ ℙ, 𝑋, 1))
5 iftrue 4531 . . . . . . . 8 (𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
65ad2antrr 724 . . . . . . 7 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
74, 6sylan9eqr 2787 . . . . . 6 ((((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑋)
8 elfznn 13557 . . . . . . . 8 (𝑋 ∈ (1...𝑁) → 𝑋 ∈ ℕ)
983ad2ant1 1130 . . . . . . 7 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → 𝑋 ∈ ℕ)
109adantl 480 . . . . . 6 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑋 ∈ ℕ)
111, 7, 10, 10fvmptd2 7006 . . . . 5 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 𝑋)
12 eleq1 2813 . . . . . . . 8 (𝑚 = 𝑌 → (𝑚 ∈ ℙ ↔ 𝑌 ∈ ℙ))
13 id 22 . . . . . . . 8 (𝑚 = 𝑌𝑚 = 𝑌)
1412, 13ifbieq1d 4549 . . . . . . 7 (𝑚 = 𝑌 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑌 ∈ ℙ, 𝑌, 1))
15 iftrue 4531 . . . . . . . 8 (𝑌 ∈ ℙ → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
1615ad2antlr 725 . . . . . . 7 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
1714, 16sylan9eqr 2787 . . . . . 6 ((((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑌)
18 elfznn 13557 . . . . . . . 8 (𝑌 ∈ (1...𝑁) → 𝑌 ∈ ℕ)
19183ad2ant2 1131 . . . . . . 7 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → 𝑌 ∈ ℕ)
2019adantl 480 . . . . . 6 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
211, 17, 20, 20fvmptd2 7006 . . . . 5 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 𝑌)
2211, 21oveq12d 7431 . . . 4 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (𝑋 gcd 𝑌))
23 prmrp 16677 . . . . . . 7 ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 gcd 𝑌) = 1 ↔ 𝑋𝑌))
2423biimprcd 249 . . . . . 6 (𝑋𝑌 → ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → (𝑋 gcd 𝑌) = 1))
25243ad2ant3 1132 . . . . 5 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → (𝑋 gcd 𝑌) = 1))
2625impcom 406 . . . 4 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝑋 gcd 𝑌) = 1)
2722, 26eqtrd 2765 . . 3 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
2827ex 411 . 2 ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
295ad2antrr 724 . . . . . . 7 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
304, 29sylan9eqr 2787 . . . . . 6 ((((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑋)
319adantl 480 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑋 ∈ ℕ)
321, 30, 31, 31fvmptd2 7006 . . . . 5 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 𝑋)
33 iffalse 4534 . . . . . . . 8 𝑌 ∈ ℙ → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
3433ad2antlr 725 . . . . . . 7 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
3514, 34sylan9eqr 2787 . . . . . 6 ((((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
3619adantl 480 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
37 1nn 12248 . . . . . . 7 1 ∈ ℕ
3837a1i 11 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 1 ∈ ℕ)
391, 35, 36, 38fvmptd2 7006 . . . . 5 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 1)
4032, 39oveq12d 7431 . . . 4 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (𝑋 gcd 1))
41 prmz 16640 . . . . . 6 (𝑋 ∈ ℙ → 𝑋 ∈ ℤ)
42 gcd1 16497 . . . . . 6 (𝑋 ∈ ℤ → (𝑋 gcd 1) = 1)
4341, 42syl 17 . . . . 5 (𝑋 ∈ ℙ → (𝑋 gcd 1) = 1)
4443ad2antrr 724 . . . 4 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝑋 gcd 1) = 1)
4540, 44eqtrd 2765 . . 3 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
4645ex 411 . 2 ((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
47 iffalse 4534 . . . . . . . 8 𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
4847ad2antrr 724 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
494, 48sylan9eqr 2787 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
509adantl 480 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑋 ∈ ℕ)
5137a1i 11 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 1 ∈ ℕ)
521, 49, 50, 51fvmptd2 7006 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 1)
5315ad2antlr 725 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
5414, 53sylan9eqr 2787 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑌)
5519adantl 480 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
561, 54, 55, 55fvmptd2 7006 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 𝑌)
5752, 56oveq12d 7431 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (1 gcd 𝑌))
58 prmz 16640 . . . . . 6 (𝑌 ∈ ℙ → 𝑌 ∈ ℤ)
59 1gcd 16503 . . . . . 6 (𝑌 ∈ ℤ → (1 gcd 𝑌) = 1)
6058, 59syl 17 . . . . 5 (𝑌 ∈ ℙ → (1 gcd 𝑌) = 1)
6160ad2antlr 725 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (1 gcd 𝑌) = 1)
6257, 61eqtrd 2765 . . 3 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
6362ex 411 . 2 ((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
6447ad2antrr 724 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
654, 64sylan9eqr 2787 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
669adantl 480 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑋 ∈ ℕ)
6737a1i 11 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 1 ∈ ℕ)
681, 65, 66, 67fvmptd2 7006 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 1)
6933ad2antlr 725 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
7014, 69sylan9eqr 2787 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
7119adantl 480 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
721, 70, 71, 67fvmptd2 7006 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 1)
7368, 72oveq12d 7431 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (1 gcd 1))
74 1z 12617 . . . . 5 1 ∈ ℤ
75 1gcd 16503 . . . . 5 (1 ∈ ℤ → (1 gcd 1) = 1)
7674, 75mp1i 13 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (1 gcd 1) = 1)
7773, 76eqtrd 2765 . . 3 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
7877ex 411 . 2 ((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
7928, 46, 63, 784cases 1038 1 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2930  ifcif 4525  cmpt 5227  cfv 6543  (class class class)co 7413  1c1 11134  cn 12237  cz 12583  ...cfz 13511   gcd cgcd 16463  cprime 16636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-z 12584  df-uz 12848  df-rp 13002  df-fz 13512  df-seq 13994  df-exp 14054  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-dvds 16226  df-gcd 16464  df-prm 16637
This theorem is referenced by:  prmodvdslcmf  17010
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