Nuprl Lemma : fps-non-trivial
∀[X:Type]. ∀[r:CRng]. ¬(1 = 0 ∈ PowerSeries(X;r)) supposing ¬(1 = 0 ∈ |r|)
Proof
Definitions occuring in Statement :
fps-one: 1,
fps-zero: 0,
power-series: PowerSeries(X;r),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
prop: ℙ,
crng: CRng,
rng: Rng,
empty-bag: {},
fps-zero: 0,
fps-coeff: f[b],
fps-one: 1,
bag-null: bag-null(bs),
null: null(as),
nil: [],
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi
Lemmas referenced :
and_wf,
equal_wf,
power-series_wf,
fps-coeff_wf,
empty-bag_wf,
fps-one_wf,
fps-zero_wf,
not_wf,
rng_car_wf,
rng_one_wf,
rng_zero_wf,
crng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
independent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
independent_pairFormation,
equalityTransitivity,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
productElimination,
voidElimination,
because_Cache,
cumulativity,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mneg{}(1 = 0) supposing \mneg{}(1 = 0)
Date html generated:
2018_05_21-PM-09_54_47
Last ObjectModification:
2017_07_26-PM-06_32_33
Theory : power!series
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