Nuprl Lemma : dot-product-split-last
∀[n:ℕ+]. ∀[x,y:ℝ^n]. (x⋅y = (x⋅y + ((x (n - 1)) * (y (n - 1)))))
Proof
Definitions occuring in Statement :
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
rmul: a * b,
radd: a + b,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat_plus: ℕ+,
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
nat: ℕ,
real-vec: ℝ^n,
sq_stable: SqStable(P),
squash: ↓T,
dot-product: x⋅y,
sq_type: SQType(T),
guard: {T},
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
so_apply: x[s]
Lemmas referenced :
dot-product-split,
nat_plus_subtype_nat,
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
istype-le,
istype-less_than,
sq_stable__req,
dot-product_wf,
radd_wf,
real-vec-subtype,
rmul_wf,
real-vec_wf,
nat_plus_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rsum_wf,
add-member-int_seg1,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
req_functionality,
req_weakening,
radd_functionality,
rsum-single,
add-zero
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
inhabitedIsType,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
productElimination,
addEquality,
setIsType,
equalityIstype,
baseApply,
sqequalBase
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (x\mcdot{}y = (x\mcdot{}y + ((x (n - 1)) * (y (n - 1)))))
Date html generated:
2019_10_30-AM-08_05_47
Last ObjectModification:
2019_07_01-AM-10_46_45
Theory : reals
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