Nuprl Lemma : dot-product-zero
∀[n:ℕ]. ∀[x:ℝ^n]. (x⋅λi.r0 = r0)
Proof
Definitions occuring in Statement :
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
dot-product: x⋅y,
real-vec: ℝ^n,
nat: ℕ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
dot-product_wf,
int-to-real_wf,
int_seg_wf,
real-vec_wf,
nat_wf,
rsum_wf,
subtract_wf,
rmul_wf,
subtract-add-cancel,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
le_wf,
req_weakening,
req_functionality,
rsum_functionality2,
rmul-zero-both,
rsum-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
independent_functionElimination,
isect_memberEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
lambdaFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. (x\mcdot{}\mlambda{}i.r0 = r0)
Date html generated:
2016_10_26-AM-10_20_50
Last ObjectModification:
2016_10_02-PM-06_29_01
Theory : reals
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