Nuprl Lemma : qexp-convex2
∀a,b:ℚ. (((0 ≤ a) ∧ (0 ≤ b)) ⇒ (∀n:ℕ+. (|a - b| ↑ n ≤ |a ↑ n - b ↑ n|)))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qabs: |r|,
qle: r ≤ s,
qsub: r - s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
qsub: r - s
Lemmas referenced :
mon_ident_q,
qinverse_q,
qadd_ac_1_q,
qadd_comm_q,
qabs-of-nonneg,
qexp-convex,
qexp_preserves_qle,
qadd_wf,
qmul_wf,
qadd_preserves_qle,
qsub_wf,
iff_weakening_equal,
qabs-difference-symmetry,
nat_wf,
true_wf,
squash_wf,
qexp_wf,
nat_plus_subtype_nat,
rationals_wf,
int-subtype-rationals,
qle_wf,
and_wf,
nat_plus_wf,
qle_connex
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
hypothesis,
isectElimination,
natural_numberEquality,
applyEquality,
sqequalRule,
because_Cache,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
rename,
minusEquality
Latex:
\mforall{}a,b:\mBbbQ{}. (((0 \mleq{} a) \mwedge{} (0 \mleq{} b)) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (|a - b| \muparrow{} n \mleq{} |a \muparrow{} n - b \muparrow{} n|)))
Date html generated:
2016_05_15-PM-11_10_48
Last ObjectModification:
2016_01_16-PM-09_24_27
Theory : rationals
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