Nuprl Lemma : geo-between-cong-tri-exists
∀[e:BasicGeometry]. ∀[a,b,c,a',c':Point].
(a_b_c ⇒ ac ≅ a'c' ⇒ (¬¬(∃b':Point. (Cong3(abc,a'b'c') ∧ Colinear(a';b';c')))))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
exists: ∃x:A. B[x],
so_apply: x[s],
basic-geometry: BasicGeometry,
and: P ∧ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
false: False,
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
geo-eq: a ≡ b,
or: P ∨ Q,
stable: Stable{P},
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
uiff: uiff(P;Q),
cand: A c∧ B,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-strict-between: a-b-c,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-between_wf,
geo-congruent_wf,
geo-colinear_wf,
geo-cong-tri_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
basic-geometry--subtype,
geo-point_wf,
exists_wf,
not_wf,
minimal-not-not-excluded-middle,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
minimal-double-negation-hyp-elim,
geo-sep_wf,
or_wf,
false_wf,
stable__not,
geo-sep_functionality,
geo-strict-between_wf,
geo-eq_wf,
lelt_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-length-flip,
geo-congruent-iff-length,
geo-congruent-between-exists,
geo-colinear-same,
geo-congruent-trivial
Rules used in proof :
isect_memberEquality,
dependent_functionElimination,
because_Cache,
rename,
setElimination,
productEquality,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
thin,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
unionElimination,
functionEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_pairFormation,
inrFormation,
inlFormation,
andLevelFunctionality,
existsFunctionality,
addLevel
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[a,b,c,a',c':Point].
(a\_b\_c {}\mRightarrow{} ac \00D0 a'c' {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}b':Point. (Cong3(abc,a'b'c') \mwedge{} Colinear(a';b';c')))))
Date html generated:
2017_10_02-PM-06_24_46
Last ObjectModification:
2017_08_05-PM-04_18_37
Theory : euclidean!plane!geometry
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