Nuprl Lemma : out-congruent
∀e:BasicGeometry. ∀a,b,c,c',c1,d,d',d1:Point.
(out(a cc') ⇒ out(b dd') ⇒ ac' ≅ bd' ⇒ a_c_c1 ⇒ b_d_d1 ⇒ cc1 ≅ bd ⇒ dd1 ≅ ac ⇒ (ac1 ≅ bd1 ∧ c'c1 ≅ d'd1))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-out: out(p ab)
Lemmas referenced :
geo-point_wf,
geo-out_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-add-length-comm,
basic-geometry_wf,
geo-length-type_wf,
true_wf,
squash_wf,
geo-add-length_wf,
geo-add-length-between,
geo-congruent-iff-length,
geo-between-sep,
geo-between-out,
geo-out_inversion,
geo-out_transitivity,
geo-out-cong-cong
Rules used in proof :
instantiate,
independent_pairFormation,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
hypothesis,
because_Cache,
independent_isectElimination,
productElimination,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,c',c1,d,d',d1:Point.
(out(a cc')
{}\mRightarrow{} out(b dd')
{}\mRightarrow{} ac' \00D0 bd'
{}\mRightarrow{} a\_c\_c1
{}\mRightarrow{} b\_d\_d1
{}\mRightarrow{} cc1 \00D0 bd
{}\mRightarrow{} dd1 \00D0 ac
{}\mRightarrow{} (ac1 \00D0 bd1 \mwedge{} c'c1 \00D0 d'd1))
Date html generated:
2017_10_02-PM-06_28_33
Last ObjectModification:
2017_08_05-PM-04_41_29
Theory : euclidean!plane!geometry
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