Nuprl Lemma : oriented-plane-axioms
∀g:OrientedPlane. ∀P:OrientedPlane ⟶ ℙ.
((∀g:OrientedPlane. (BasicGeometryAxioms(g) ⇒ geo-left-axioms(g) ⇒ P[g])) ⇒ P[g])
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-left-axioms: geo-left-axioms(g),
basic-geo-axioms: BasicGeometryAxioms(g),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
and: P ∧ Q,
oriented-plane: Error :oriented-plane,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-left-axioms_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
subtype_rel_transitivity,
Error :oriented-plane-subtype,
basic-geometry--subtype,
basic-geo-axioms_wf,
Error :oriented-plane_wf,
all_wf,
sq_stable__geo-left-axioms-1,
Error :o-geo-structure-subtype,
sq_stable__geo-axioms
Rules used in proof :
universeEquality,
functionExtensionality,
independent_isectElimination,
functionEquality,
cumulativity,
lambdaEquality,
isectElimination,
instantiate,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
applyEquality,
productElimination,
extract_by_obid,
introduction,
rename,
setElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:OrientedPlane. \mforall{}P:OrientedPlane {}\mrightarrow{} \mBbbP{}.
((\mforall{}g:OrientedPlane. (BasicGeometryAxioms(g) {}\mRightarrow{} geo-left-axioms(g) {}\mRightarrow{} P[g])) {}\mRightarrow{} P[g])
Date html generated:
2017_10_02-PM-06_49_07
Last ObjectModification:
2017_08_06-PM-07_28_47
Theory : euclidean!plane!geometry
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