Nuprl Lemma : rv-pos-angle-permute
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ rv-pos-angle(n;c;a;b))
Proof
Definitions occuring in Statement :
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-pos-angle: rv-pos-angle(n;a;b;c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
nat: ℕ,
real-vec: ℝ^n,
uimplies: b supposing a,
rsub: x - y,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-vec-mul: a*X,
subtype_rel: A ⊆r B,
true: True,
absval: |i|,
squash: ↓T
Lemmas referenced :
rv-pos-angle-permute-lemma,
real-vec-sub_wf,
rv-pos-angle_wf,
real-vec_wf,
nat_wf,
int_seg_wf,
req_wf,
rsub_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
rminus_wf,
req_weakening,
rabs_wf,
dot-product_wf,
real-vec-norm_wf,
uiff_transitivity,
req_functionality,
radd_functionality,
rminus-radd,
req_inversion,
radd-assoc,
req_transitivity,
radd-ac,
radd_comm,
rmul_functionality,
rminus-as-rmul,
rminus-rminus,
radd-rminus-assoc,
rless_functionality,
rabs_functionality,
dot-product_functionality,
req-vec_weakening,
real-vec-norm_functionality,
real-vec-norm-diff,
rmul_comm,
equal_wf,
real-vec-mul_wf,
dot-product-linearity2,
absval_wf,
rabs-rmul,
squash_wf,
true_wf,
real_wf,
rabs-int,
dot-product-comm,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
independent_functionElimination,
sqequalRule,
natural_numberEquality,
setElimination,
rename,
applyEquality,
because_Cache,
minusEquality,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (rv-pos-angle(n;a;b;c) {}\mRightarrow{} rv-pos-angle(n;c;a;b))
Date html generated:
2017_10_03-AM-10_57_58
Last ObjectModification:
2017_03_02-AM-11_08_34
Theory : reals
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