Nuprl Lemma : rat-sub-cube_transitivity
∀[k:ℕ]. ∀a,b,c:ℚCube(k). (rat-sub-cube(k;a;b) ⇒ rat-sub-cube(k;b;c) ⇒ rat-sub-cube(k;a;c))
Proof
Definitions occuring in Statement :
rat-sub-cube: rat-sub-cube(k;a;b),
rational-cube: ℚCube(k),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
rat-sub-cube: rat-sub-cube(k;a;b),
member: t ∈ T,
rational-cube: ℚCube(k),
rational-interval: ℚInterval,
rat-sub-interval: rat-sub-interval(I;J),
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
nat: ℕ
Lemmas referenced :
qle_transitivity_qorder,
qless_wf,
qle_wf,
int_seg_wf,
rat-sub-cube_wf,
rational-cube_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
inhabitedIsType,
productElimination,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
universeIsType,
productIsType,
functionIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
setElimination,
rename
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}a,b,c:\mBbbQ{}Cube(k). (rat-sub-cube(k;a;b) {}\mRightarrow{} rat-sub-cube(k;b;c) {}\mRightarrow{} rat-sub-cube(k;a;c))
Date html generated:
2020_05_20-AM-09_17_49
Last ObjectModification:
2019_11_14-PM-10_56_18
Theory : rationals
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