Nuprl Lemma : r2-det-convex2
∀[p,q,r,t,s:ℝ^2]. ∀[a,b,c:ℝ].
|a*p + b*q + c*rts| = ((a * |pts|) + (b * |qts|) + (c * |rts|)) supposing ((a + b + c) = r1) ∧ r1 - a ≠ r0
Proof
Definitions occuring in Statement :
r2-det: |pqr|,
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec: ℝ^n,
rneq: x ≠ y,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
req-vec: req-vec(n;x;y),
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec: ℝ^n,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rsub: x - y
Lemmas referenced :
req_witness,
r2-det_wf,
real-vec-add_wf,
false_wf,
le_wf,
real-vec-mul_wf,
radd_wf,
rmul_wf,
req_wf,
int-to-real_wf,
rneq_wf,
rsub_wf,
real_wf,
real-vec_wf,
r2-det-convex1,
rdiv_wf,
int_seg_wf,
equal_wf,
req_weakening,
uiff_transitivity,
req_functionality,
rmul-distrib,
radd_functionality,
rmul_functionality,
rmul_comm,
rmul-ac,
rmul-rdiv-cancel,
r2-det_functionality,
real-vec-add_functionality,
req-vec_weakening,
rmul_preserves_req,
rminus_wf,
radd-preserves-req,
squash_wf,
true_wf,
iff_weakening_equal,
rmul-rdiv-cancel2,
req_transitivity,
rmul_over_rminus,
rmul-one-both,
rminus_functionality,
radd_comm,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-zero-both,
radd-zero-both,
radd-ac,
radd-rminus-both,
real-vec-mul_functionality,
rmul-assoc,
radd-rminus-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
because_Cache,
independent_functionElimination,
productEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
minusEquality,
addEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[p,q,r,t,s:\mBbbR{}\^{}2]. \mforall{}[a,b,c:\mBbbR{}].
|a*p + b*q + c*rts| = ((a * |pts|) + (b * |qts|) + (c * |rts|))
supposing ((a + b + c) = r1) \mwedge{} r1 - a \mneq{} r0
Date html generated:
2017_10_03-AM-11_45_17
Last ObjectModification:
2017_04_11-PM-05_32_29
Theory : reals
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