Nuprl Lemma : isosceles-mid-cong-angles
∀e:HeytingGeometry. ∀a,b,c,m:Point. (c # ab ⇒ a=m=b ⇒ ac ≅ bc ⇒ acm ≅a bcm)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-cong-angle: abc ≅a xyz,
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
heyting-geometry: HeytingGeometry,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
uiff: uiff(P;Q),
geo-midpoint: a=m=b,
exists: ∃x:A. B[x],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
geo-sep-sym,
geo-triangle-property,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-midpoint_wf,
geo-triangle_wf,
geo-point_wf,
geo-between-trivial,
geo-congruent-iff-length,
euclidean-plane-subtype-basic,
basic-geometry_wf,
geo-length-flip,
geo-congruent-refl,
geo-between_wf,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
midpoint-sep,
geo-triangle-colinear,
geo-triangle-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
productElimination,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productIsType,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
lambdaEquality_alt
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,m:Point. (c \# ab {}\mRightarrow{} a=m=b {}\mRightarrow{} ac \mcong{} bc {}\mRightarrow{} acm \mcong{}\msuba{} bcm)
Date html generated:
2019_10_16-PM-02_09_19
Last ObjectModification:
2018_12_14-AM-10_00_27
Theory : euclidean!plane!geometry
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