Nuprl Lemma : cong-angle-out-aux2
∀g:HeytingGeometry. ∀a,b,c,d,e,f,a',c',d',f':Point.
((a # bc ∧ a'c' ≅ d'f')
⇒ d # ef
⇒ out(b a'a)
⇒ out(b c'c)
⇒ out(e d'd)
⇒ out(e f'f)
⇒ ba' ≅ ed'
⇒ bc' ≅ ef'
⇒ abc ≅a def)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
geo-cong-angle: abc ≅a xyz,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
heyting-geometry: HeytingGeometry,
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
geo-out: out(p ab),
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-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-out_wf,
euclidean-plane-subtype-basic,
basic-geometry_wf,
geo-triangle_wf,
geo-point_wf,
geo-proper-extend-exists,
geo-sep-sym,
geo-strict-between-implies-between,
subtype_rel_self,
basic-geometry-_wf,
geo-between-symmetry,
geo-congruent-iff-length,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-add-length-comm,
geo-between_wf,
geo-between-out,
geo-strict-between-sep1,
geo-out_transitivity,
geo-out_inversion,
geo-out-cong-cong,
geo-colinear-five-segment,
geo-colinear-is-colinear-set,
geo-out-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,
geo-length-flip
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
productIsType,
inhabitedIsType,
rename,
dependent_pairFormation_alt,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation
Latex:
\mforall{}g:HeytingGeometry. \mforall{}a,b,c,d,e,f,a',c',d',f':Point.
((a \# bc \mwedge{} a'c' \mcong{} d'f')
{}\mRightarrow{} d \# ef
{}\mRightarrow{} out(b a'a)
{}\mRightarrow{} out(b c'c)
{}\mRightarrow{} out(e d'd)
{}\mRightarrow{} out(e f'f)
{}\mRightarrow{} ba' \mcong{} ed'
{}\mRightarrow{} bc' \mcong{} ef'
{}\mRightarrow{} abc \mcong{}\msuba{} def)
Date html generated:
2019_10_16-PM-02_08_17
Last ObjectModification:
2018_12_15-PM-09_45_25
Theory : euclidean!plane!geometry
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