Nuprl Lemma : real-vec-mul-cancel
∀[n:ℕ]. ∀[v:ℝ^n]. ∀[x,y:ℝ]. (v ≠ λi.r0 ⇒ x = y supposing req-vec(n;x*v;y*v))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
req-vec: req-vec(n;x;y),
real-vec-mul: a*X,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
req_witness,
req-vec_wf,
real-vec-mul_wf,
real-vec-sep_wf,
int-to-real_wf,
int_seg_wf,
real_wf,
real-vec_wf,
nat_wf,
real-vec-sep-iff-rneq,
rmul_preserves_req,
req_functionality,
rmul_wf,
rmul_comm,
req-implies-req,
rsub_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productElimination,
applyEquality,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[v:\mBbbR{}\^{}n]. \mforall{}[x,y:\mBbbR{}]. (v \mneq{} \mlambda{}i.r0 {}\mRightarrow{} x = y supposing req-vec(n;x*v;y*v))
Date html generated:
2018_05_22-PM-02_26_45
Last ObjectModification:
2018_03_23-AM-11_05_36
Theory : reals
Home
Index