Nuprl Lemma : rdiv-int-fractions
∀a,b:ℤ. ∀c,d:ℕ+. ((r(a)/r(c))/(r(b)/r(d))) = (r(a * d)/r(c * b)) supposing ¬(b = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
nat_plus: ℕ+,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T
Lemmas referenced :
rmul_preserves_rneq_iff2,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
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,
not_wf,
equal-wf-base,
int_subtype_base,
nat_plus_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
rmul-one,
rmul-zero-both,
rneq_wf,
rneq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
int_entire_a,
real_wf,
equal_wf,
rmul_preserves_req,
req_witness,
req_functionality,
req_inversion,
rmul-int,
rmul_assoc,
rinv-mul-as-rdiv,
rmul-rinv3,
req_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation,
productElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyEquality,
baseClosed,
addLevel,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}a,b:\mBbbZ{}. \mforall{}c,d:\mBbbN{}\msupplus{}. ((r(a)/r(c))/(r(b)/r(d))) = (r(a * d)/r(c * b)) supposing \mneg{}(b = 0)
Date html generated:
2018_05_22-PM-01_33_12
Last ObjectModification:
2017_10_22-PM-03_46_26
Theory : reals
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