Nuprl Lemma : r2-straightedge-compass-ext
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2| b ≠ a ∧ c_b_d} .
∃u:{u:ℝ^2| cu=cd ∧ a_b_u}
(∃v:ℝ^2 [(cv=cd ∧ v_b_u ∧ (¬((¬a_b_v) ∧ (¬b_v_a) ∧ (¬v_a_b))) ∧ (b ≠ d ⇒ (v ≠ u ∧ u ≠ b ∧ v ≠ b)))])
Proof
Definitions occuring in Statement :
rv-be: a_b_c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
let: let,
rvlinecircle0: rvlinecircle0(n;a;b;p;q),
r2-straightedge-compass,
rv-extend,
rv-extend-1,
rv-line-circle-3-ext,
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x]
Lemmas referenced :
set_wf,
real-vec-sep_wf,
rv-be_wf,
real-vec_wf,
false_wf,
le_wf,
r2-straightedge-compass,
subtype_rel_self,
all_wf,
exists_wf,
rv-congruent_wf,
sq_exists_wf,
not_wf,
rv-extend,
rv-extend-1,
rv-line-circle-3-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
applyEquality,
instantiate,
functionEquality,
setEquality,
setElimination,
rename,
productElimination
Latex:
\mforall{}c,d,a:\mBbbR{}\^{}2. \mforall{}b:\{b:\mBbbR{}\^{}2| b \mneq{} a \mwedge{} c\_b\_d\} .
\mexists{}u:\{u:\mBbbR{}\^{}2| cu=cd \mwedge{} a\_b\_u\}
(\mexists{}v:\mBbbR{}\^{}2 [(cv=cd
\mwedge{} v\_b\_u
\mwedge{} (\mneg{}((\mneg{}a\_b\_v) \mwedge{} (\mneg{}b\_v\_a) \mwedge{} (\mneg{}v\_a\_b)))
\mwedge{} (b \mneq{} d {}\mRightarrow{} (v \mneq{} u \mwedge{} u \mneq{} b \mwedge{} v \mneq{} b)))])
Date html generated:
2018_05_22-PM-02_37_15
Last ObjectModification:
2018_05_18-AM-09_48_28
Theory : reals
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