Nuprl Lemma : geo-between-outer-trans-cpy
∀e:EuclideanPlane. ∀[a,b,c,d:Point]. (a_c_d) supposing (b_c_d and a_b_c and b ≠ c)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
not: ¬A,
false: False,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf,
stable__geo-between,
false_wf,
not_wf,
extend-using-SC,
geo-construction-unicity2,
geo-between-symmetry,
geo-between-inner-trans,
geo-congruent-symmetry,
istype-void,
geo-between-trivial2,
minimal-double-negation-hyp-elim,
geo-between_functionality,
geo-eq_weakening,
minimal-not-not-excluded-middle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
setElimination,
rename,
unionEquality,
functionEquality,
independent_functionElimination,
productElimination,
functionIsType,
unionIsType,
unionElimination,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[a,b,c,d:Point]. (a\_c\_d) supposing (b\_c\_d and a\_b\_c and b \mneq{} c)
Date html generated:
2019_10_16-PM-01_15_33
Last ObjectModification:
2019_01_17-PM-03_20_22
Theory : euclidean!plane!geometry
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