Nuprl Lemma : qinv-wf
∀[r:ℤ ⋃ (ℤ × ℤ-o)]. 1/r ∈ ℤ ⋃ (ℤ × ℤ-o) supposing ¬↑qeq(r;0)
Proof
Definitions occuring in Statement :
qinv: 1/r,
qeq: qeq(r;s),
int_nzero: ℤ-o,
b-union: A ⋃ B,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
qinv: 1/r,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
btrue: tt,
bfalse: ff,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
qeq: qeq(r;s),
evalall: evalall(t),
eq_int: (i =z j),
assert: ↑b,
true: True,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
int_nzero_properties,
assert_of_eq_int,
set-valueall-type,
product-valueall-type,
ifthenelse_wf,
nequal_wf,
equal_wf,
int_subtype_base,
subtype_base_sq,
evalall-reduce,
int-valueall-type,
valueall-type-has-valueall,
b-union_wf,
int_nzero_wf,
subtype_rel_b-union-left,
qeq_wf,
assert_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
intEquality,
productEquality,
isect_memberEquality,
because_Cache,
imageElimination,
productElimination,
unionElimination,
equalityElimination,
independent_isectElimination,
callbyvalueReduce,
isintReduceTrue,
imageMemberEquality,
dependent_pairEquality,
independent_pairEquality,
dependent_set_memberEquality,
lambdaFormation,
independent_functionElimination,
instantiate,
cumulativity,
dependent_functionElimination,
voidElimination,
universeEquality,
lambdaEquality,
sqleReflexivity,
multiplyEquality,
setElimination,
rename,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll
Latex:
\mforall{}[r:\mBbbZ{} \mcup{} (\mBbbZ{} \mtimes{} \mBbbZ{}\msupminus{}\msupzero{})]. 1/r \mmember{} \mBbbZ{} \mcup{} (\mBbbZ{} \mtimes{} \mBbbZ{}\msupminus{}\msupzero{}) supposing \mneg{}\muparrow{}qeq(r;0)
Date html generated:
2016_05_15-PM-10_38_06
Last ObjectModification:
2016_01_16-PM-09_37_10
Theory : rationals
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