Nuprl Lemma : sum_of_geometric_prog_q
∀[a:ℚ]. ∀[n:ℕ]. (((1 + -(a)) * Σ0 ≤ i < n. a ↑ i) = (1 + -(a ↑ n)) ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qsum: Σa ≤ j < b. E[j],
qmul: r * s,
qadd: r + s,
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
q-rng-nexp: q-rng-nexp(r;n),
qsum: Σa ≤ j < b. E[j],
uall: ∀[x:A]. B[x],
member: t ∈ T,
qrng: <ℚ+*>,
rng_car: |r|,
pi1: fst(t),
rng_times: *,
pi2: snd(t),
rng_plus: +r,
rng_one: 1,
rng_minus: -r,
infix_ap: x f y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
squash: ↓T,
true: True,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
nat: ℕ,
subtype_rel: A ⊆r B
Lemmas referenced :
sum_of_geometric_prog,
qrng_wf,
iff_weakening_equal,
qexp-eq-q-rng-nexp,
qsum_wf,
true_wf,
squash_wf,
equal_wf,
int_seg_wf,
false_wf,
int_seg_subtype_nat,
qmul_wf,
int-subtype-rationals,
qadd_wf,
rationals_wf,
nat_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
intEquality,
functionEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
rename,
setElimination,
minusEquality,
applyEquality,
natural_numberEquality,
because_Cache,
hypothesisEquality,
isect_memberFormation
Latex:
\mforall{}[a:\mBbbQ{}]. \mforall{}[n:\mBbbN{}]. (((1 + -(a)) * \mSigma{}0 \mleq{} i < n. a \muparrow{} i) = (1 + -(a \muparrow{} n)))
Date html generated:
2020_05_20-AM-09_25_57
Last ObjectModification:
2020_02_03-PM-02_28_16
Theory : rationals
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