Nuprl Lemma : eu-seg-extend_functionality
∀e:EuclideanPlane. ∀[s1,s2:ProperSegment]. ∀[t1,t2:Segment]. (s1 + t1 ≡ s2 + t2) supposing (t1 ≡ t2 and s1 ≡ s2)
Proof
Definitions occuring in Statement :
eu-seg-extend: s + t,
eu-seg-congruent: s1 ≡ s2,
eu-proper-segment: ProperSegment,
eu-segment: Segment,
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
subtype_rel: A ⊆r B,
eu-proper-segment: ProperSegment,
sq_stable: SqStable(P),
implies: P ⇒ Q,
euclidean-axioms: euclidean-axioms(e),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
not: ¬A,
false: False,
eu-seg-extend: s + t,
eu-seg-congruent: s1 ≡ s2,
eu-seg2: s.2,
eu-seg1: s.1,
pi1: fst(t),
pi2: snd(t),
eu-seg-proper: proper(s),
eu-congruent: ab=cd,
record-select: r.x
Lemmas referenced :
eu-congruent-symmetry,
eu-congruent-transitivity,
eu-three-segment,
eu-congruent_wf,
eu-between-eq_wf,
and_wf,
eu-extend_wf,
not_wf,
eu-seg2_wf,
eu-seg1_wf,
eu-point_wf,
equal_wf,
euclidean-plane_wf,
eu-segment_wf,
eu-seg-congruent_wf,
eu-proper-segment_wf,
eu-seg-extend_wf,
sq_stable_eu-seg-congruent
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lemma_by_obid,
dependent_functionElimination,
isectElimination,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
independent_functionElimination,
introduction,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
because_Cache,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[s1,s2:ProperSegment]. \mforall{}[t1,t2:Segment]. (s1 + t1 \mequiv{} s2 + t2) supposing (t1 \mequiv{} t2 and s1 \mequiv{} s2)
Date html generated:
2016_05_18-AM-06_37_10
Last ObjectModification:
2016_01_16-PM-10_31_56
Theory : euclidean!geometry
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