Nuprl Lemma : int_part_decomp_wf
∀[q:ℚ]. (int_part_decomp(q) ∈ {p:ℤ × ℚ| (0 ≤ (snd(p))) ∧ snd(p) < 1 ∧ (q = ((fst(p)) + (snd(p))) ∈ ℚ)} )
Proof
Definitions occuring in Statement :
int_part_decomp: int_part_decomp(q),
qle: r ≤ s,
qless: r < s,
qadd: r + s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_part_decomp: int_part_decomp(q),
all: ∀x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
pi2: snd(t),
pi1: fst(t),
cand: A c∧ B
Lemmas referenced :
rat-int-part_wf2,
set_wf,
rationals_wf,
qle_wf,
qless_wf,
equal_wf,
qadd_wf,
int-subtype-rationals
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
productEquality,
intEquality,
setEquality,
natural_numberEquality,
applyEquality,
because_Cache,
lambdaEquality,
spreadEquality,
productElimination,
independent_pairEquality,
setElimination,
rename,
dependent_set_memberEquality,
lambdaFormation,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[q:\mBbbQ{}]. (int\_part\_decomp(q) \mmember{} \{p:\mBbbZ{} \mtimes{} \mBbbQ{}| (0 \mleq{} (snd(p))) \mwedge{} snd(p) < 1 \mwedge{} (q = ((fst(p)) + (snd(p))))\} )
Date html generated:
2018_05_22-AM-00_30_35
Last ObjectModification:
2017_07_26-PM-06_58_26
Theory : rationals
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