Nuprl Lemma : r2-circle-circle
∀a,b,c,d:ℝ^2. ∀p:{p:ℝ^2| ab=ap} . ∀q:{q:ℝ^2| cd=cq} . ∀x:{x:ℝ^2| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^2| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c ⇒ (∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp} . ((x ≠ d ∧ y ≠ b) ⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
all: ∀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,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
nat: ℕ,
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
false: False,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
rv-circle-circle-lemma3',
real-vec-sep_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
r2-left_wf,
rv-congruent_wf,
rv-between_wf,
real-vec_wf,
rv-non-strict-between-iff,
rv-Tsep,
real-vec-sep-symmetry,
rv-between-symmetry,
real-vec-dist-be,
real-vec-dist_wf,
radd_wf,
radd-preserves-rless,
rminus_wf,
int-to-real_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
rless_functionality,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
sqequalRule,
independent_pairFormation,
productIsType,
universeIsType,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
because_Cache,
functionIsType,
setIsType,
inhabitedIsType,
applyEquality,
equalityTransitivity,
equalitySymmetry,
int_eqEquality
Latex:
\mforall{}a,b,c,d:\mBbbR{}\^{}2. \mforall{}p:\{p:\mBbbR{}\^{}2| ab=ap\} . \mforall{}q:\{q:\mBbbR{}\^{}2| cd=cq\} . \mforall{}x:\{x:\mBbbR{}\^{}2|
cp=cx \mwedge{} (\mneg{}(c \mneq{} x \mwedge{} x \mneq{} d \mwedge{} (\mneg{}c-x-d)))\} .
\mforall{}y:\{y:\mBbbR{}\^{}2| aq=ay \mwedge{} (\mneg{}(a \mneq{} y \mwedge{} y \mneq{} b \mwedge{} (\mneg{}a-y-b)))\} .
(a \mneq{} c {}\mRightarrow{} (\mexists{}u,v:\{p:\mBbbR{}\^{}2| ab=ap \mwedge{} cd=cp\} . ((x \mneq{} d \mwedge{} y \mneq{} b) {}\mRightarrow{} (r2-left(u;c;a) \mwedge{} r2-left(v;a;c)))))
Date html generated:
2019_10_30-AM-08_55_53
Last ObjectModification:
2018_12_11-AM-10_54_26
Theory : reals
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