Nuprl Lemma : implies-rv-pos-angle
∀n:ℕ. ∀a,b,c,a':ℝ^n. (a-b-a' ⇒ ab=a'b ⇒ ab=cb ⇒ c ≠ a ⇒ c ≠ a' ⇒ rv-pos-angle(n;a;b;c))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
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],
rv-congruent: ab=cd,
real-vec-norm: ||x||,
rsqrt: rsqrt(x),
rroot: rroot(i;x),
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
btrue: tt,
rroot-abs: rroot-abs(i;x),
fastexp: i^n,
efficient-exp-ext,
subtract: n - m,
real-vec-dist: d(x;y),
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
real-vec-sep: a ≠ b,
rv-between: a-b-c,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
real-vec-mul: a*X,
real-vec-add: X + Y,
nat: ℕ,
real-vec: ℝ^n,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
less_than: a < b,
squash: ↓T,
true: True,
rdiv: (x/y),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T)
Lemmas referenced :
rv-pos-angle-lemma,
real-vec-sub_wf,
real-vec-sep_wf,
rv-congruent_wf,
rv-between_wf,
real-vec_wf,
istype-nat,
int-to-real_wf,
real-vec-dist_wf,
rless_functionality,
req_weakening,
real-vec-dist-translation,
real-vec-sep-symmetry,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
req_functionality,
real-vec-dist_functionality,
req-vec_weakening,
req-vec_functionality,
real-vec-mul_functionality,
real-vec-sub_functionality,
real-vec-dist-equal-iff,
int_seg_wf,
radd_wf,
rmul_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
dot-product_wf,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
dot-product_functionality,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rnexp_wf,
istype-false,
istype-le,
real-vec-norm_wf,
rnexp-positive,
req_inversion,
real-vec-norm-squared,
rless_wf,
req_wf,
iff_weakening_uiff,
rmul-assoc,
req_transitivity,
dot-product-linearity2,
rmul_functionality,
rmul_preserves_req,
radd-preserves-req,
rdiv_wf,
rless-int,
rinv_wf2,
rmul-rinv,
minus-one-mul-top,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rsub_functionality,
radd_functionality,
int-rinv-cancel2,
real-vec-add_functionality,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
rmul-rinv3,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
independent_functionElimination,
sqequalRule,
universeIsType,
inhabitedIsType,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
productElimination,
minusEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityIsType1,
inrFormation_alt,
closedConclusion,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
intEquality,
equalityIsType4,
imageElimination,
universeEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,a':\mBbbR{}\^{}n. (a-b-a' {}\mRightarrow{} ab=a'b {}\mRightarrow{} ab=cb {}\mRightarrow{} c \mneq{} a {}\mRightarrow{} c \mneq{} a' {}\mRightarrow{} rv-pos-angle(n;a;b;c))
Date html generated:
2019_10_30-AM-08_48_29
Last ObjectModification:
2018_11_08-PM-02_13_53
Theory : reals
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