Nuprl Lemma : common-face-inhabited-intersection
∀k:ℕ. ∀a,b,b':ℚCube(k). ((↑Inhabited(b')) ⇒ (↑Inhabited(b)) ⇒ (a ≤ b ∧ a ≤ b') ⇒ (↑Inhabited(b ⋂ b')))
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
rat-cube-intersection: c ⋂ d,
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
rat-cube-intersection: c ⋂ d,
rat-cube-face: c ≤ d,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
rational-cube: ℚCube(k),
rational-interval: ℚInterval,
rat-interval-intersection: I ⋂ J,
inhabited-rat-interval: Inhabited(I),
rat-interval-face: I ≤ J,
rat-point-interval: [a],
or: P ∨ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
pi2: snd(t),
pi1: fst(t),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
rational-interval_wf,
qle_wf,
assert-q_le-eq,
iff_weakening_equal,
qmax_wf,
qmin_wf,
subtype_rel_self,
istype-assert,
q_le_wf,
int_seg_wf,
rat-interval-face_wf,
inhabited-rat-interval_wf,
rational-cube_wf,
istype-nat,
iff_weakening_uiff,
assert_wf,
inhabited-rat-cube_wf,
assert-inhabited-rat-cube,
rat-cube-intersection_wf,
rat-cube-face_wf,
pi2_wf,
rationals_wf,
pi1_wf_top,
istype-void,
qmax_lb,
qle_transitivity_qorder,
qle_weakening_eq_qorder,
qmin_ub
Rules used in proof :
cut,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
applyEquality,
inhabitedIsType,
unionIsType,
equalityIstype,
universeIsType,
introduction,
extract_by_obid,
independent_pairEquality,
isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
productIsType,
functionIsType,
functionEquality,
imageElimination,
unionElimination,
applyLambdaEquality,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productEquality,
promote_hyp
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b,b':\mBbbQ{}Cube(k).
((\muparrow{}Inhabited(b')) {}\mRightarrow{} (\muparrow{}Inhabited(b)) {}\mRightarrow{} (a \mleq{} b \mwedge{} a \mleq{} b') {}\mRightarrow{} (\muparrow{}Inhabited(b \mcap{} b')))
Date html generated:
2020_05_20-AM-09_20_21
Last ObjectModification:
2019_11_01-PM-11_06_49
Theory : rationals
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