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Theorem fvprmselgcd1 17079
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 2827 . . . . . . . 8 (𝑚 = 𝑋 → (𝑚 ∈ ℙ ↔ 𝑋 ∈ ℙ))
3 id 22 . . . . . . . 8 (𝑚 = 𝑋𝑚 = 𝑋)
42, 3ifbieq1d 4555 . . . . . . 7 (𝑚 = 𝑋 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑋 ∈ ℙ, 𝑋, 1))
5 iftrue 4537 . . . . . . . 8 (𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
65ad2antrr 726 . . . . . . 7 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
74, 6sylan9eqr 2797 . . . . . 6 ((((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑋)
8 elfznn 13590 . . . . . . . 8 (𝑋 ∈ (1...𝑁) → 𝑋 ∈ ℕ)
983ad2ant1 1132 . . . . . . 7 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → 𝑋 ∈ ℕ)
109adantl 481 . . . . . 6 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑋 ∈ ℕ)
111, 7, 10, 10fvmptd2 7024 . . . . 5 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 𝑋)
12 eleq1 2827 . . . . . . . 8 (𝑚 = 𝑌 → (𝑚 ∈ ℙ ↔ 𝑌 ∈ ℙ))
13 id 22 . . . . . . . 8 (𝑚 = 𝑌𝑚 = 𝑌)
1412, 13ifbieq1d 4555 . . . . . . 7 (𝑚 = 𝑌 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑌 ∈ ℙ, 𝑌, 1))
15 iftrue 4537 . . . . . . . 8 (𝑌 ∈ ℙ → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
1615ad2antlr 727 . . . . . . 7 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
1714, 16sylan9eqr 2797 . . . . . 6 ((((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑌)
18 elfznn 13590 . . . . . . . 8 (𝑌 ∈ (1...𝑁) → 𝑌 ∈ ℕ)
19183ad2ant2 1133 . . . . . . 7 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → 𝑌 ∈ ℕ)
2019adantl 481 . . . . . 6 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
211, 17, 20, 20fvmptd2 7024 . . . . 5 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 𝑌)
2211, 21oveq12d 7449 . . . 4 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (𝑋 gcd 𝑌))
23 prmrp 16746 . . . . . . 7 ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 gcd 𝑌) = 1 ↔ 𝑋𝑌))
2423biimprcd 250 . . . . . 6 (𝑋𝑌 → ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → (𝑋 gcd 𝑌) = 1))
25243ad2ant3 1134 . . . . 5 ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → (𝑋 gcd 𝑌) = 1))
2625impcom 407 . . . 4 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝑋 gcd 𝑌) = 1)
2722, 26eqtrd 2775 . . 3 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
2827ex 412 . 2 ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
295ad2antrr 726 . . . . . . 7 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
304, 29sylan9eqr 2797 . . . . . 6 ((((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑋)
319adantl 481 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑋 ∈ ℕ)
321, 30, 31, 31fvmptd2 7024 . . . . 5 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 𝑋)
33 iffalse 4540 . . . . . . . 8 𝑌 ∈ ℙ → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
3433ad2antlr 727 . . . . . . 7 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
3514, 34sylan9eqr 2797 . . . . . 6 ((((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
3619adantl 481 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
37 1nn 12275 . . . . . . 7 1 ∈ ℕ
3837a1i 11 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 1 ∈ ℕ)
391, 35, 36, 38fvmptd2 7024 . . . . 5 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 1)
4032, 39oveq12d 7449 . . . 4 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (𝑋 gcd 1))
41 prmz 16709 . . . . . 6 (𝑋 ∈ ℙ → 𝑋 ∈ ℤ)
42 gcd1 16562 . . . . . 6 (𝑋 ∈ ℤ → (𝑋 gcd 1) = 1)
4341, 42syl 17 . . . . 5 (𝑋 ∈ ℙ → (𝑋 gcd 1) = 1)
4443ad2antrr 726 . . . 4 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝑋 gcd 1) = 1)
4540, 44eqtrd 2775 . . 3 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
4645ex 412 . 2 ((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
47 iffalse 4540 . . . . . . . 8 𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
4847ad2antrr 726 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
494, 48sylan9eqr 2797 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
509adantl 481 . . . . . 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 7024 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 1)
5315ad2antlr 727 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
5414, 53sylan9eqr 2797 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑌)
5519adantl 481 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
561, 54, 55, 55fvmptd2 7024 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 𝑌)
5752, 56oveq12d 7449 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (1 gcd 𝑌))
58 prmz 16709 . . . . . 6 (𝑌 ∈ ℙ → 𝑌 ∈ ℤ)
59 1gcd 16567 . . . . . 6 (𝑌 ∈ ℤ → (1 gcd 𝑌) = 1)
6058, 59syl 17 . . . . 5 (𝑌 ∈ ℙ → (1 gcd 𝑌) = 1)
6160ad2antlr 727 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (1 gcd 𝑌) = 1)
6257, 61eqtrd 2775 . . 3 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
6362ex 412 . 2 ((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
6447ad2antrr 726 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
654, 64sylan9eqr 2797 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
669adantl 481 . . . . . 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 7024 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 1)
6933ad2antlr 727 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
7014, 69sylan9eqr 2797 . . . . . 6 ((((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1)
7119adantl 481 . . . . . 6 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 𝑌 ∈ ℕ)
721, 70, 71, 67fvmptd2 7024 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 1)
7368, 72oveq12d 7449 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (1 gcd 1))
74 1z 12645 . . . . 5 1 ∈ ℤ
75 1gcd 16567 . . . . 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 2775 . . 3 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
7877ex 412 . 2 ((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) → ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1))
7928, 46, 63, 784cases 1040 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 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  ifcif 4531  cmpt 5231  cfv 6563  (class class class)co 7431  1c1 11154  cn 12264  cz 12611  ...cfz 13544   gcd cgcd 16528  cprime 16705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529  df-prm 16706
This theorem is referenced by:  prmodvdslcmf  17081
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