Nuprl Lemma : r2-det-convex1
∀[p,q,r,t:ℝ^2]. ∀[a:ℝ]. (|a*p + r1 - a*tqr| = ((a * |pqr|) + ((r1 - a) * |tqr|)))
Proof
Definitions occuring in Statement :
r2-det: |pqr|,
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec: ℝ^n,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
all: ∀x:A. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermAdd: left (+) right,
itermMultiply: left (*) right,
itermVar: vvar,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
real_wf,
real-vec_wf,
false_wf,
le_wf,
r2-det_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
lelt_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
r2-det-add,
radd_functionality,
req_weakening,
r2-det-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesisEquality,
applyEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
productElimination,
independent_isectElimination
Latex:
\mforall{}[p,q,r,t:\mBbbR{}\^{}2]. \mforall{}[a:\mBbbR{}]. (|a*p + r1 - a*tqr| = ((a * |pqr|) + ((r1 - a) * |tqr|)))
Date html generated:
2017_10_03-AM-11_44_50
Last ObjectModification:
2017_04_11-PM-05_32_05
Theory : reals
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