Nuprl Lemma : eu-cong3-to-conga-aux
∀e:EuclideanPlane. ∀b,a,a',a0,e0,d,d',d0:Point.
(out(b aa') ⇒ out(e0 dd') ⇒ b_a_a0 ⇒ e0_d_d0 ⇒ ba'=e0d' ⇒ aa0=e0d ⇒ dd0=ba ⇒ (ba0=e0d0 ∧ a'a0=d'd0))
Proof
Definitions occuring in Statement :
eu-out: out(p ab),
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-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,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
not: ¬A,
uiff: uiff(P;Q),
eu-out: out(p ab),
stable: Stable{P},
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True
Lemmas referenced :
eu-congruent_wf,
eu-between-eq_wf,
eu-out_wf,
eu-point_wf,
euclidean-plane_wf,
eu-five-segment,
equal_wf,
eu-between-eq-symmetry,
eu-congruent-iff-length,
eu-length-flip,
eu-congruent-trivial,
stable__eu-congruent,
not_wf,
eu-between-eq-same-side,
eu-between-eq-seg-eq,
eu-add-length-between,
set_wf,
eu-O_wf,
eu-X_wf,
iff_weakening_equal,
and_wf,
eu-add-length_wf,
eu-length_wf,
eu-mk-seg_wf,
eu-add-length-zero,
eu-add-length-assoc,
eu-add-length-cancel-left,
eu-between-eq-trivial-right,
eu-sum-eq-x,
eu-ab-eq-x,
squash_wf,
true_wf,
euclidean-structure_wf,
eu-segment_wf,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityEquality,
sqequalRule,
lambdaEquality,
dependent_set_memberEquality,
setEquality,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
hyp_replacement,
Error :applyLambdaEquality,
equalityElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}b,a,a',a0,e0,d,d',d0:Point.
(out(b aa')
{}\mRightarrow{} out(e0 dd')
{}\mRightarrow{} b\_a\_a0
{}\mRightarrow{} e0\_d\_d0
{}\mRightarrow{} ba'=e0d'
{}\mRightarrow{} aa0=e0d
{}\mRightarrow{} dd0=ba
{}\mRightarrow{} (ba0=e0d0 \mwedge{} a'a0=d'd0))
Date html generated:
2016_10_26-AM-07_45_17
Last ObjectModification:
2016_07_12-AM-08_49_46
Theory : euclidean!geometry
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