Nuprl Lemma : sup-angles-preserve-congruence
∀e:HeytingGeometry. ∀a,b,c,x,y,z,a',x':Point. ((abc ≅a xyz ∧ a # bc ∧ x # yz) ⇒ a-b-a' ⇒ x-y-x' ⇒ a'bc ≅a x'yz)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-cong-angle: abc ≅a xyz,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
heyting-geometry: Error :heyting-geometry,
true: True,
prop: ℙ,
squash: ↓T,
uiff: uiff(P;Q),
geo-strict-between: a-b-c,
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
subtract: n - m,
cons: [a / b],
select: L[n],
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
cand: A c∧ B
Lemmas referenced :
geo-point_wf,
Error :geo-triangle_wf,
geo-cong-angle_wf,
geo-strict-between_wf,
geo-add-length-comm,
geo-length-type_wf,
true_wf,
squash_wf,
geo-add-length_wf,
geo-congruent-iff-length,
geo-add-length-between,
geo-between-inner-trans,
geo-strict-between-sep1,
geo-between-out,
geo-out_inversion,
geo-between-outer-trans,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-bet-out-out-bet,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-strict-between-sep3,
geo-proper-extend-exists,
heyting-geometry-subtype,
cong-angle-out-exists3,
geo-length-flip,
geo-sep-sym,
geo-five-segment,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-triangle-property,
geo-strict-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-triangle-symmetry,
geo-triangle-colinear,
cong-angle-out-aux2
Rules used in proof :
setElimination,
productEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
rename,
independent_isectElimination,
isectElimination,
instantiate,
because_Cache,
independent_functionElimination,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,x,y,z,a',x':Point.
((abc \00D0\msuba{} xyz \mwedge{} a \# bc \mwedge{} x \# yz) {}\mRightarrow{} a-b-a' {}\mRightarrow{} x-y-x' {}\mRightarrow{} a'bc \00D0\msuba{} x'yz)
Date html generated:
2017_10_02-PM-07_04_13
Last ObjectModification:
2017_08_08-PM-00_36_16
Theory : euclidean!plane!geometry
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