Nuprl Lemma : ip-inner-Pasch'
∀rv:InnerProductSpace. ∀a,b:Point. ∀c:{c:Point| b # c} . ∀p:{p:Point| a_p_c ∧ a # p} . ∀q:{q:Point| b_q_c} .
(∃x:{Point| (a_x_q
∧ b_x_p
∧ (a # q ⇒ x # a)
∧ ((a # q ∧ p # c ∧ b # q) ⇒ x # q)
∧ ((b # p ∧ b # q) ⇒ x # b)
∧ ((b # p ∧ q # c) ⇒ x # p))})
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
ip-inner-Pasch,
ip-between_wf,
sq_stable__rv-sep-ext,
sq_stable__ip-between,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
set_wf,
ss-point_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
isectElimination,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberFormation,
independent_pairFormation,
because_Cache,
productEquality,
functionEquality,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b:Point. \mforall{}c:\{c:Point| b \# c\} . \mforall{}p:\{p:Point| a\_p\_c \mwedge{} a \# p\} . \mforall{}q:\{q:Point|
b\_q\_c\} .
(\mexists{}x:\{Point| (a\_x\_q
\mwedge{} b\_x\_p
\mwedge{} (a \# q {}\mRightarrow{} x \# a)
\mwedge{} ((a \# q \mwedge{} p \# c \mwedge{} b \# q) {}\mRightarrow{} x \# q)
\mwedge{} ((b \# p \mwedge{} b \# q) {}\mRightarrow{} x \# b)
\mwedge{} ((b \# p \mwedge{} q \# c) {}\mRightarrow{} x \# p))\})
Date html generated:
2017_10_05-AM-00_05_12
Last ObjectModification:
2017_03_15-PM-09_01_27
Theory : inner!product!spaces
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