Nuprl Lemma : qmul_wf
∀[r,s:ℚ]. (r * s ∈ ℚ)
Proof
Definitions occuring in Statement :
qmul: r * s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rationals: ℚ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
qmul: r * s,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
btrue: tt,
qeq: qeq(r;s),
bfalse: ff,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
rationals_wf,
quotient-member-eq,
b-union_wf,
int_nzero_wf,
equal-wf-T-base,
bool_wf,
qeq_wf,
qeq-equiv,
valueall-type-has-valueall,
bunion-valueall-type,
int-valueall-type,
product-valueall-type,
istype-int,
set-valueall-type,
nequal_wf,
evalall-reduce,
isint-int,
eqtt_to_assert,
eq_int_wf,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
btrue_wf,
subtype_rel_b-union-left,
subtype_rel_b-union-right,
trivial-equal,
bfalse_wf,
int_entire_a,
int_nzero_properties,
full-omega-unsat,
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,
ifthenelse_wf,
mul-commutes,
mul-swap,
mul-associates,
mul_preserves_eq,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
thin,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
isectElimination,
intEquality,
productEquality,
lambdaEquality_alt,
hypothesisEquality,
baseClosed,
independent_isectElimination,
dependent_functionElimination,
imageElimination,
unionElimination,
equalityElimination,
independent_functionElimination,
because_Cache,
natural_numberEquality,
callbyvalueReduce,
isintReduceTrue,
multiplyEquality,
independent_pairEquality,
Error :memTop,
setElimination,
instantiate,
cumulativity,
equalityIstype,
productIsType,
sqequalBase,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
applyEquality,
imageMemberEquality,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
universeEquality
Latex:
\mforall{}[r,s:\mBbbQ{}]. (r * s \mmember{} \mBbbQ{})
Date html generated:
2020_05_20-AM-09_12_55
Last ObjectModification:
2019_12_31-PM-08_25_37
Theory : rationals
Home
Index