Nuprl Lemma : rat2real-qdiv
∀a:ℚ. ∀b:ℤ-o. (rat2real((a/b)) = (rat2real(a)/r(b)))
Proof
Definitions occuring in Statement :
rat2real: rat2real(q),
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
int_nzero: ℤ-o,
all: ∀x:A. B[x],
qdiv: (r/s),
rationals: ℚ
Definitions unfolded in proof :
or: P ∨ Q,
rneq: x ≠ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
btrue: tt,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
qinv: 1/r,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T ,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
prop: ℙ,
qmul: r * s,
bfalse: ff,
ifthenelse: if b then t else f fi ,
qdiv: (r/s),
rat2real: rat2real(q),
mk-rational: mk-rational(a;b),
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
not: ¬A,
cand: A c∧ B,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rmul-rinv3,
rinv-of-rmul,
rinv_functionality2,
rneq_functionality,
int-rdiv-req,
rless_wf,
rless-int,
int_entire_a,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rmul-rinv,
req_weakening,
rmul-int,
req_inversion,
int-rdiv_functionality,
rmul_functionality,
req_transitivity,
req_functionality,
rinv_wf2,
itermMultiply_wf,
itermSubtract_wf,
rmul_wf,
int-rdiv_wf,
rmul_preserves_req,
int_formula_prop_less_lemma,
intformless_wf,
mul_nzero,
evalall-sqequal,
evalall-reduce,
set-valueall-type,
int-valueall-type,
product-valueall-type,
valueall-type-has-valueall,
int_nzero_wf,
set_subtype_base,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformnot_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
int_nzero_properties,
rneq-int,
int-to-real_wf,
rdiv_wf,
int_nzero-rational,
nequal_wf,
subtype_rel_set,
qdiv_wf,
rat2real_wf,
req_wf,
mk-rational-qdiv,
istype-assert,
assert-qeq,
int_subtype_base,
rationals_wf,
equal-wf-base,
int-subtype-rationals,
qeq_wf2,
assert_wf,
iff_weakening_uiff,
nat_plus_properties,
q-elim
Rules used in proof :
inrFormation_alt,
equalityTransitivity,
multiplyEquality,
dependent_set_memberEquality_alt,
closedConclusion,
baseApply,
isintReduceTrue,
callbyvalueReduce,
independent_pairEquality,
productEquality,
sqequalBase,
equalityIstype,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
inhabitedIsType,
lambdaEquality_alt,
intEquality,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
baseClosed,
because_Cache,
natural_numberEquality,
sqequalRule,
applyEquality,
independent_functionElimination,
rename,
setElimination,
hypothesis,
isectElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbQ{}. \mforall{}b:\mBbbZ{}\msupminus{}\msupzero{}. (rat2real((a/b)) = (rat2real(a)/r(b)))
Date html generated:
2019_10_31-AM-05_56_53
Last ObjectModification:
2019_10_30-PM-02_48_48
Theory : reals
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