Nuprl Lemma : perp-in-congruence
∀e:EuclideanPlane
∀[a,b,A,B,c,d,C,D:Point].
(ad ≅ AD ∧ bd ≅ BD ∧ cd ≅ CD) supposing (ab ⊥d dc and AB ⊥D DC and bc ≅ BC and ac ≅ AC and ab ≅ AB)
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q
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,
implies: P ⇒ Q,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry: BasicGeometry,
geo-perp-in: ab ⊥x cd,
and: P ∧ Q,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
stable: Stable{P},
geo-cong-tri: Cong3(abc,a'b'c'),
geo-perp: ab ⊥ cd,
cand: A c∧ B,
oriented-plane: OrientedPlane,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
geo-out: out(p ab),
geo-eq: a ≡ b,
geo-strict-between: a-b-c
Lemmas referenced :
sq_stable__and,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
sq_stable__geo-congruent,
geo-perp-in_wf,
geo-lsep_wf,
geo-point_wf,
geo-colinear-cong-tri-exists,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
stable__and,
stable__geo-congruent,
false_wf,
or_wf,
exists_wf,
geo-cong-tri_wf,
geo-colinear_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-perp-unicity,
lsep-implies-sep,
geo-perp-in-symmetry,
geo-perp-in-iff2,
lsep-colinear-sep,
geo-sep_wf,
geo-colinear-symmetry,
geo-sep-sym,
stable__right-angle,
geo-cong-angle_wf,
right-angle_wf,
geo-sas2,
geo-congruent-iff-length,
geo-length-flip,
congruence-preserves-right-angle,
geo-colinear-same,
cong-tri-implies-cong-angle,
geo-colinear-out-cases,
geo-out_wf,
geo-between_wf,
geo-congruent-preserves-out,
geo-congruent-full-symmetry,
lsep-all-sym,
geo-out-trivial,
out-cong-angle,
geo-cong-angle-symmetry,
euclidean-plane-axioms,
geo-cong-angle-symm2,
geo-cong-angle-transitivity,
geo-congruent-preserves-between,
geo-between-symmetry,
stable__not,
geo-between_functionality,
geo-eq_weakening,
geo-colinear_functionality,
geo-cong-angle_functionality,
geo-congruent_functionality,
geo-perp-in_functionality,
geo-lsep_functionality,
supplementary-angles-preserve-congruence,
geo-cong-angle-symm3,
geo-congruent-symmetry,
geo-congruent-sep,
right-angle-trivial2,
geo-congruence-identity,
right-angle_functionality,
geo-eq_inversion,
congruence-preserves-lsep,
not-lsep-iff-colinear,
geo-congruent-preserves-colinear,
geo-perp-trivial-when-colinear,
geo-perp-in-not-eq,
geo-colinear-five-segment
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
isect_memberEquality_alt,
instantiate,
independent_isectElimination,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
productIsType,
lambdaEquality_alt,
productEquality,
inhabitedIsType,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
isectIsType,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[a,b,A,B,c,d,C,D:Point].
(ad \mcong{} AD \mwedge{} bd \mcong{} BD \mwedge{} cd \mcong{} CD) supposing
(ab \mbot{}d dc and
AB \mbot{}D DC and
bc \mcong{} BC and
ac \mcong{} AC and
ab \mcong{} AB)
Date html generated:
2019_10_16-PM-01_58_40
Last ObjectModification:
2018_11_07-PM-01_08_04
Theory : euclidean!plane!geometry
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