Nuprl Lemma : r2-straightedge-compass
∀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,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
real-vec: ℝ^n,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
real-vec-sub: X - Y,
nat_plus: ℕ+,
less_than: a < b,
true: True,
real: ℝ,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
real-vec-sep: a ≠ b,
rge: x ≥ y,
guard: {T},
rv-congruent: ab=cd,
rv-be: a_b_c,
rv-between: a-b-c,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
set_wf,
real-vec_wf,
false_wf,
le_wf,
real-vec-sep_wf,
rv-be_wf,
sq_stable__real-vec-sep,
real_wf,
ifthenelse_wf,
eq_int_wf,
radd_wf,
rabs_wf,
int-to-real_wf,
int_seg_wf,
req_wf,
real-vec-dist_wf,
exists_wf,
rleq_wf,
dot-product_wf,
real-vec-sub_wf,
rmul_wf,
real-regular,
rsub_wf,
less_than_wf,
regular-int-seq_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
r2-dot-product,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rsqrt_wf,
dot-product-nonneg,
square-nonneg,
rsqrt-of-square,
radd-non-neg,
zero-rleq-rabs,
rleq-int,
rsqrt_functionality,
rv-extend,
real-vec-between_wf,
rless_wf,
rless_functionality,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
sq_stable__rless,
real-vec-dist-between,
req_transitivity,
radd_functionality,
real-vec-dist-nonneg,
rabs-bounds,
radd-preserves-rless,
rminus_wf,
radd-rminus-both,
itermMinus_wf,
real_term_value_minus_lemma,
rv-be-symmetry,
not_wf,
rv-between_wf,
rless_transitivity2,
real-vec-triangle-inequality,
rless_transitivity1,
real-vec-dist-symmetry,
rless-implies-rless,
rv-line-circle-3-ext,
sq_stable__rv-be,
rv-be-inner-trans,
req_inversion,
rv-congruent_wf,
sq_exists_wf,
rv-pos-angle_wf,
rv-be-dist,
rv-pos-angle-shift,
rv-pos-angle-permute,
rv-pos-angle-not-be,
rv-pos-angle-symmetry,
not-rv-pos-angle-implies2,
rv-T-iff,
rv-T_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
hypothesisEquality,
lambdaEquality,
productEquality,
because_Cache,
setElimination,
rename,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
applyEquality,
setEquality,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
impliesFunctionality,
promote_hyp
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_36_58
Last ObjectModification:
2018_05_18-AM-09_46_36
Theory : reals
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