Nuprl Lemma : integer-fractional-parts
∀q:ℚ. (q = (integer-part(q) + fractional-part(q)) ∈ ℚ)
Proof
Definitions occuring in Statement :
fractional-part: fractional-part(q),
integer-part: integer-part(q),
qadd: r + s,
rationals: ℚ,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
fractional-part: fractional-part(q),
integer-part: integer-part(q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
pi2: snd(t),
prop: ℙ,
pi1: fst(t),
so_apply: x[s],
implies: P ⇒ Q
Lemmas referenced :
int_part_decomp_wf,
set_wf,
rationals_wf,
qle_wf,
qless_wf,
equal_wf,
qadd_wf,
int-subtype-rationals
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productEquality,
intEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
because_Cache,
productElimination,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}q:\mBbbQ{}. (q = (integer-part(q) + fractional-part(q)))
Date html generated:
2018_05_22-AM-00_30_56
Last ObjectModification:
2017_07_26-PM-06_58_46
Theory : rationals
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