Nuprl Lemma : radd-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℕ+]. (((r(a)/r(c)) + (r(b)/r(d))) = (r((a * d) + (b * c))/r(c * d)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
req: x = y,
radd: a + b,
int-to-real: r(n),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
req_witness,
radd_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
mul_bounds_1b,
nat_plus_wf,
rmul_wf,
rneq_functionality,
rmul-int,
req_weakening,
rmul_preserves_req,
req_wf,
rinv_wf2,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
uiff_transitivity,
req_functionality,
rmul_functionality,
rdiv_functionality,
req_transitivity,
req_inversion,
radd-int,
radd_functionality,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addEquality,
multiplyEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[c,d:\mBbbN{}\msupplus{}]. (((r(a)/r(c)) + (r(b)/r(d))) = (r((a * d) + (b * c))/r(c * d)))
Date html generated:
2017_10_03-AM-08_38_00
Last ObjectModification:
2017_07_28-AM-07_30_25
Theory : reals
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