Nuprl Lemma : ip-inner-Pasch
∀rv:InnerProductSpace. ∀a,b,c:Point. ∀p:{p:Point| a_p_c} . ∀q:{q:Point| b_q_c} .
(a # p
⇒ b # c
⇒ (∃x:{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],
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,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
ip-inner-Pasch1,
sq_stable__ip-between,
ip-between_wf,
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,
independent_functionElimination,
isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
functionEquality,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
lambdaEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. \mforall{}p:\{p:Point| a\_p\_c\} . \mforall{}q:\{q:Point| b\_q\_c\} .
(a \# p
{}\mRightarrow{} b \# c
{}\mRightarrow{} (\mexists{}x:\{x:Point| a\_x\_q \mwedge{} b\_x\_p\}
((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_08
Last ObjectModification:
2017_03_15-AM-10_28_45
Theory : inner!product!spaces
Home
Index