Nuprl Lemma : real-vec-acute-angle
∀n:ℕ. ∀x,y,z:ℝ^n. (r0 < x - y⋅z - y ⇐⇒ ∀x':ℝ^n. ((d(x';y) = d(x;y)) ⇒ real-vec-be(n;x;y;x') ⇒ (d(z;x) < d(z;x'))))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
dot-product: x⋅y,
real-vec-be: real-vec-be(n;a;b;c),
real-vec-sub: X - Y,
real-vec: ℝ^n,
rless: x < y,
req: x = y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
req-vec: req-vec(n;x;y),
real-vec-mul: a*X,
real-vec-sub: X - Y,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
real-vec-be: real-vec-be(n;a;b;c),
exists: ∃x:A. B[x],
or: P ∨ Q,
stable: Stable{P},
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
i-member: r ∈ I,
rccint: [l, u],
top: Top,
guard: {T},
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rneq: x ≠ y,
rdiv: (x/y),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
real-vec-add: X + Y,
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
decidable: Dec(P),
pi2: snd(t),
pi1: fst(t),
rtermSubtract: left "-" right,
rtermVar: rtermVar(var),
rtermConstant: "const",
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermAdd: left "+" right,
rat_term_to_real: rat_term_to_real(f;t)
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y,z:\mBbbR{}\^{}n.
(r0 < x - y\mcdot{}z - y
\mLeftarrow{}{}\mRightarrow{} \mforall{}x':\mBbbR{}\^{}n. ((d(x';y) = d(x;y)) {}\mRightarrow{} real-vec-be(n;x;y;x') {}\mRightarrow{} (d(z;x) < d(z;x'))))
Date html generated:
2020_05_20-PM-00_49_19
Last ObjectModification:
2019_12_14-PM-03_01_28
Theory : reals
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