Nuprl Lemma : congruence-preserves-right-angle2
∀e:BasicGeometry. ∀a,b,c:Point. (Rabc ⇒ (∀a',b',c':Point. (abc ≅a a'b'c' ⇒ Ra'b'c')))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
right-angle: Rabc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
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 :
cong-angle-out-exists-cong3,
geo-cong-angle-symm2,
congruence-preserves-right-angle,
geo-congruent-symmetry,
geo-cong-angle_wf,
right-angle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
right-angle-colinear,
geo-sep-sym,
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,
right-angle-colinear2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
independent_pairFormation,
because_Cache,
isectElimination,
independent_isectElimination,
universeIsType,
inhabitedIsType,
applyEquality,
instantiate,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (Rabc {}\mRightarrow{} (\mforall{}a',b',c':Point. (abc \mcong{}\msuba{} a'b'c' {}\mRightarrow{} Ra'b'c')))
Date html generated:
2019_10_16-PM-01_50_25
Last ObjectModification:
2019_06_07-PM-04_10_01
Theory : euclidean!plane!geometry
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