Nuprl Lemma : rmul-int-rdiv
∀[x:ℝ]. ∀[a,b:ℤ]. ((r(a) * (r(b)/x)) = (r(a * b)/x)) supposing x ≠ r0
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rmul-int,
rmul-rdiv-cancel2,
rmul-rdiv-cancel,
rmul_comm,
rmul_functionality,
rmul-ac,
req_transitivity,
rmul-assoc,
req_inversion,
req_functionality,
uiff_transitivity,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
req-int,
req_weakening,
req_wf,
real_wf,
rneq_wf,
req_witness,
rdiv_wf,
int-to-real_wf,
rmul_wf,
rmul_preserves_req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
because_Cache,
multiplyEquality,
productElimination,
independent_functionElimination,
intEquality,
sqequalRule,
isect_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[a,b:\mBbbZ{}]. ((r(a) * (r(b)/x)) = (r(a * b)/x)) supposing x \mneq{} r0
Date html generated:
2016_05_18-AM-07_27_53
Last ObjectModification:
2016_01_17-AM-01_58_17
Theory : reals
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