primes.divides_to_mod_O
∀ n m, O < n ⇒ n | m ⇒ (mod m n) = O
Theorem divides_to_mod_O : forall (n:nat.nat), forall (m:nat.nat), (nat.lt nat.O n) -> (divides n m) -> logic.eq (nat.nat) (div_mod.mod m n) nat.O.
theorem divides_to_mod_O : \forall (n:nat). \forall (m:nat). ((lt) (O) n) -> ((divides) n m) -> (eq) (nat) ((mod) m n) (O) .
theorem divides_to_mod_O : forall (n:nat.nat) , forall (m:nat.nat) , ((((nat.lt_) ) ((nat.O) )) (n)) -> ((((primes.divides) ) (n)) (m)) -> (((logic.eq_) (nat.nat)) ((((div_mod.mod) ) (m)) (n))) ((nat.O) ).
divides_to_mod_O : LEMMA (FORALL(n:nat_sttfa_th.sttfa_nat):(FORALL(m:nat_sttfa_th.sttfa_nat):(nat_sttfa_th.lt(nat_sttfa_th.sttfa_O)(n) => (primes_sttfa.sttfa_divides(n)(m) => logic_sttfa_th.eq[nat_sttfa_th.sttfa_nat](div_mod_sttfa_th.mod(m)(n))(nat_sttfa_th.sttfa_O)))))
Printing for OpenTheory is not working at the moment.