Nuprl Lemma : int-rinv-cancel
∀[a:ℤ]. ∀[b:ℤ-o]. ∀[x:ℝ]. ((r(a * b) * rinv(r(b)) * x) = (r(a) * x))
Proof
Definitions occuring in Statement :
rinv: rinv(x),
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
int-to-real_wf,
rinv_wf2,
rneq-int,
int_nzero_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-T-base,
real_wf,
int_nzero_wf,
req_wf,
rmul-one,
req_weakening,
rmul_functionality,
req_functionality,
req_inversion,
rmul_assoc,
uiff_transitivity,
rmul-int,
rmul-rinv1,
rnonzero-iff,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
multiplyEquality,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
productElimination,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
baseClosed
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. \mforall{}[x:\mBbbR{}]. ((r(a * b) * rinv(r(b)) * x) = (r(a) * x))
Date html generated:
2017_10_03-AM-08_34_28
Last ObjectModification:
2017_04_05-PM-04_22_08
Theory : reals
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