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### Theoremgcd.divides_mod_to_divides

Statement

∀ p m n, O < n ⇒ p | (mod m n) ⇒ p | n ⇒ p | m

Main Dependencies

Statement

Theorem divides_mod_to_divides : forall (p:nat.nat), forall (m:nat.nat), forall (n:nat.nat), (nat.lt nat.O n) -> (primes.divides p (div_mod.mod m n)) -> (primes.divides p n) -> primes.divides p m.

Statement

theorem divides_mod_to_divides : \forall (p:nat). \forall (m:nat). \forall (n:nat). ((lt) (O) n) -> ((divides) p ((mod) m n)) -> ((divides) p n) -> (divides) p m.

Statement

theorem divides_mod_to_divides : forall (p:nat.nat) , forall (m:nat.nat) , forall (n:nat.nat) , ((((nat.lt_) ) ((nat.O) )) (n)) -> ((((primes.divides) ) (p)) ((((div_mod.mod) ) (m)) (n))) -> ((((primes.divides) ) (p)) (n)) -> (((primes.divides) ) (p)) (m).

Statement

divides_mod_to_divides : LEMMA (FORALL(p:nat_sttfa_th.sttfa_nat):(FORALL(m:nat_sttfa_th.sttfa_nat):(FORALL(n:nat_sttfa_th.sttfa_nat):(nat_sttfa_th.lt(nat_sttfa_th.sttfa_O)(n) => (primes_sttfa_th.sttfa_divides(p)(div_mod_sttfa_th.mod(m)(n)) => (primes_sttfa_th.sttfa_divides(p)(n) => primes_sttfa_th.sttfa_divides(p)(m)))))))

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