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Theorem fvprmselgcd1 16999
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 2816 . . . . . . . 8 (𝑚 = 𝑋 → (𝑚 ∈ ℙ ↔ 𝑋 ∈ ℙ))
3 id 22 . . . . . . . 8 (𝑚 = 𝑋𝑚 = 𝑋)
42, 3ifbieq1d 4548 . . . . . . 7 (𝑚 = 𝑋 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑋 ∈ ℙ, 𝑋, 1))
5 iftrue 4530 . . . . . . . 8 (𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
65ad2antrr 725 . . . . . . 7 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
74, 6sylan9eqr 2789 . . . . . 6 ((((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑋)
8 elfznn 13548 . . . . . . . 8 (𝑋 ∈ (1...𝑁) → 𝑋 ∈ ℕ)
983ad2ant1 1131 . . . . . . 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 7007 . . . . 5 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 𝑋)
12 eleq1 2816 . . . . . . . 8 (𝑚 = 𝑌 → (𝑚 ∈ ℙ ↔ 𝑌 ∈ ℙ))
13 id 22 . . . . . . . 8 (𝑚 = 𝑌𝑚 = 𝑌)
1412, 13ifbieq1d 4548 . . . . . . 7 (𝑚 = 𝑌 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑌 ∈ ℙ, 𝑌, 1))
15 iftrue 4530 . . . . . . . 8 (𝑌 ∈ ℙ → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
1615ad2antlr 726 . . . . . . 7 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
1714, 16sylan9eqr 2789 . . . . . 6 ((((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) ∧ 𝑚 = 𝑌) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑌)
18 elfznn 13548 . . . . . . . 8 (𝑌 ∈ (1...𝑁) → 𝑌 ∈ ℕ)
19183ad2ant2 1132 . . . . . . 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 7007 . . . . 5 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 𝑌)
2211, 21oveq12d 7432 . . . 4 (((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (𝑋 gcd 𝑌))
23 prmrp 16668 . . . . . . 7 ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → ((𝑋 gcd 𝑌) = 1 ↔ 𝑋𝑌))
2423biimprcd 249 . . . . . 6 (𝑋𝑌 → ((𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) → (𝑋 gcd 𝑌) = 1))
25243ad2ant3 1133 . . . . 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 2767 . . 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 725 . . . . . . 7 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋)
304, 29sylan9eqr 2789 . . . . . 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 7007 . . . . 5 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 𝑋)
33 iffalse 4533 . . . . . . . 8 𝑌 ∈ ℙ → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
3433ad2antlr 726 . . . . . . 7 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
3514, 34sylan9eqr 2789 . . . . . 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 12239 . . . . . . 7 1 ∈ ℕ
3837a1i 11 . . . . . 6 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → 1 ∈ ℕ)
391, 35, 36, 38fvmptd2 7007 . . . . 5 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 1)
4032, 39oveq12d 7432 . . . 4 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (𝑋 gcd 1))
41 prmz 16631 . . . . . 6 (𝑋 ∈ ℙ → 𝑋 ∈ ℤ)
42 gcd1 16488 . . . . . 6 (𝑋 ∈ ℤ → (𝑋 gcd 1) = 1)
4341, 42syl 17 . . . . 5 (𝑋 ∈ ℙ → (𝑋 gcd 1) = 1)
4443ad2antrr 725 . . . 4 (((𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝑋 gcd 1) = 1)
4540, 44eqtrd 2767 . . 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 4533 . . . . . . . 8 𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
4847ad2antrr 725 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
494, 48sylan9eqr 2789 . . . . . 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 7007 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 1)
5315ad2antlr 726 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 𝑌)
5414, 53sylan9eqr 2789 . . . . . 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 7007 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 𝑌)
5752, 56oveq12d 7432 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (1 gcd 𝑌))
58 prmz 16631 . . . . . 6 (𝑌 ∈ ℙ → 𝑌 ∈ ℤ)
59 1gcd 16494 . . . . . 6 (𝑌 ∈ ℤ → (1 gcd 𝑌) = 1)
6058, 59syl 17 . . . . 5 (𝑌 ∈ ℙ → (1 gcd 𝑌) = 1)
6160ad2antlr 726 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (1 gcd 𝑌) = 1)
6257, 61eqtrd 2767 . . 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 725 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1)
654, 64sylan9eqr 2789 . . . . . 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 7007 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑋) = 1)
6933ad2antlr 726 . . . . . . 7 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → if(𝑌 ∈ ℙ, 𝑌, 1) = 1)
7014, 69sylan9eqr 2789 . . . . . 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 7007 . . . . 5 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → (𝐹𝑌) = 1)
7368, 72oveq12d 7432 . . . 4 (((¬ 𝑋 ∈ ℙ ∧ ¬ 𝑌 ∈ ℙ) ∧ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌)) → ((𝐹𝑋) gcd (𝐹𝑌)) = (1 gcd 1))
74 1z 12608 . . . . 5 1 ∈ ℤ
75 1gcd 16494 . . . . 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 2767 . . 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 1039 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 1085   = wceq 1534  wcel 2099  wne 2935  ifcif 4524  cmpt 5225  cfv 6542  (class class class)co 7414  1c1 11125  cn 12228  cz 12574  ...cfz 13502   gcd cgcd 16454  cprime 16627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-inf 9452  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-fz 13503  df-seq 13985  df-exp 14045  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201  df-dvds 16217  df-gcd 16455  df-prm 16628
This theorem is referenced by:  prmodvdslcmf  17001
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