Nuprl Lemma : dot-product-linearity1
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. ((x + y ⋅ z = (x ⋅ z + y ⋅ z)) ∧ (z ⋅ x + y = (z ⋅ x + z ⋅ y)))
Proof
Definitions occuring in Statement :
dot-product: x ⋅ y,
real-vec-add: X + Y,
real-vec: ℝ^n,
req: x = y,
radd: a + b,
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
dot-product: x ⋅ y,
real-vec-add: X + Y,
real-vec: ℝ^n,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
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,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
so_apply: x[s],
pointwise-req: x[k] = y[k] for k ∈ [n,m],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rsum_linearity1,
req_inversion,
req_weakening,
req_functionality,
rmul-distrib1,
le_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
itermSubtract_wf,
intformle_wf,
rmul-distrib2,
rsum_functionality,
nat_wf,
real_wf,
int_seg_wf,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
subtract-add-cancel,
radd_wf,
rmul_wf,
subtract_wf,
rsum_wf,
req_witness
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lemma_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
addEquality,
independent_functionElimination,
functionEquality,
lambdaFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y,z:\mBbbR{}\^{}n]. ((x + y \mcdot{} z = (x \mcdot{} z + y \mcdot{} z)) \mwedge{} (z \mcdot{} x + y = (z \mcdot{} x + z \mcdot{} y)))
Date html generated:
2016_05_18-AM-09_47_34
Last ObjectModification:
2016_01_17-AM-02_51_22
Theory : reals
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