Nuprl Lemma : face-of-intersection
∀k:ℕ. ∀a,b,b':ℚCube(k). ((↑Inhabited(b')) ⇒ (↑Inhabited(b)) ⇒ (a ≤ b ∧ a ≤ b') ⇒ a ≤ 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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
uimplies: b supposing a,
rat-cube-face: c ≤ d,
rat-cube-intersection: c ⋂ d,
cand: A c∧ B,
rational-cube: ℚCube(k),
nat: ℕ,
prop: ℙ,
guard: {T},
rational-interval: ℚInterval,
rat-interval-face: I ≤ J,
rat-interval-intersection: I ⋂ J,
rat-point-interval: [a],
inhabited-rat-interval: Inhabited(I),
or: P ∨ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
pi2: snd(t),
pi1: fst(t),
true: True,
rev_implies: P ⇐ Q,
squash: ↓T
Lemmas referenced :
assert-inhabited-rat-cube,
int_seg_wf,
rat-cube-face_wf,
istype-assert,
inhabited-rat-cube_wf,
rational-cube_wf,
istype-nat,
rational-interval_wf,
qle_wf,
assert-q_le-eq,
iff_weakening_equal,
q_le_wf,
pi2_wf,
rationals_wf,
pi1_wf_top,
istype-void,
equal_wf,
or_wf,
qmax_wf,
qmin_wf,
subtype_rel_self,
squash_wf,
true_wf,
istype-universe,
qmax-idempotent,
qmin-idempotent,
qmax-eq-iff-1,
or_assoc,
member_wf,
qmax-eq-iff-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
independent_pairFormation,
applyEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
natural_numberEquality,
setElimination,
rename,
productIsType,
unionIsType,
independent_pairEquality,
because_Cache,
promote_hyp,
unionElimination,
applyLambdaEquality,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
hyp_replacement,
inlFormation_alt,
inrFormation_alt,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed
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{} a \mleq{} b \mcap{} b')
Date html generated:
2020_05_20-AM-09_20_29
Last ObjectModification:
2019_11_02-AM-10_59_05
Theory : rationals
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