Nuprl Lemma : case-real2_wf2
∀[d,a,b:ℝ]. ∀[f:a ≠ b ⟶ ((↓r0 < d) ∨ (↓¬(r0 < d)))].
(case-real2(a;b;f) ∈ {z:ℝ| ((r0 < d) ⇒ (z = a)) ∧ ((d ≤ r0) ⇒ (z = b))} )
Proof
Definitions occuring in Statement :
case-real2: case-real2(a;b;f),
rneq: x ≠ y,
rleq: x ≤ y,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
guard: {T},
uimplies: b supposing a,
false: False,
prop: ℙ,
or: P ∨ Q
Lemmas referenced :
case-real2_wf,
rless_wf,
int-to-real_wf,
rless_transitivity1,
rless_irreflexivity,
rleq_wf,
req_wf,
rneq_wf,
squash_wf,
not_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
lambdaFormation_alt,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
voidElimination,
universeIsType,
sqequalRule,
productIsType,
functionIsType,
inhabitedIsType,
unionIsType
Latex:
\mforall{}[d,a,b:\mBbbR{}]. \mforall{}[f:a \mneq{} b {}\mrightarrow{} ((\mdownarrow{}r0 < d) \mvee{} (\mdownarrow{}\mneg{}(r0 < d)))].
(case-real2(a;b;f) \mmember{} \{z:\mBbbR{}| ((r0 < d) {}\mRightarrow{} (z = a)) \mwedge{} ((d \mleq{} r0) {}\mRightarrow{} (z = b))\} )
Date html generated:
2019_10_29-AM-09_37_07
Last ObjectModification:
2019_05_23-PM-05_57_53
Theory : reals
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