Nuprl Lemma : req-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℤ-o]. uiff((r(a)/r(c)) = (r(b)/r(d));(a * d) = (b * c) ∈ ℤ)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
int_nzero: ℤ-o,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
int_nzero: ℤ-o,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_wf,
rdiv_wf,
int-to-real_wf,
rneq-int,
int_nzero_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
set_subtype_base,
nequal_wf,
int_subtype_base,
req_witness,
int_nzero_wf,
rmul_preserves_req,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
req_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul-int,
req-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
productElimination,
independent_functionElimination,
lambdaFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
equalityIstype,
inhabitedIsType,
applyEquality,
intEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
baseApply,
closedConclusion,
independent_pairEquality,
axiomEquality,
isectIsTypeImplies,
multiplyEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[c,d:\mBbbZ{}\msupminus{}\msupzero{}]. uiff((r(a)/r(c)) = (r(b)/r(d));(a * d) = (b * c))
Date html generated:
2019_10_29-AM-09_58_26
Last ObjectModification:
2019_01_10-PM-00_20_48
Theory : reals
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