Nuprl Lemma : r2-dot-product
∀[a,b:ℝ^2]. (a⋅b = (((a 0) * (b 0)) + ((a 1) * (b 1))))
Proof
Definitions occuring in Statement :
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
rmul: a * b,
radd: a + b,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
dot-product: x⋅y,
subtract: n - m,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
so_lambda: λ2x.t[x],
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],
top: Top,
so_apply: x[s],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
dot-product_wf,
false_wf,
le_wf,
radd_wf,
rmul_wf,
lelt_wf,
real-vec_wf,
rsum_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
equal-wf-base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
req_weakening,
req_functionality,
rsum-split-first,
radd_functionality,
rsum-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality,
lambdaEquality,
setElimination,
rename,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
setEquality
Latex:
\mforall{}[a,b:\mBbbR{}\^{}2]. (a\mcdot{}b = (((a 0) * (b 0)) + ((a 1) * (b 1))))
Date html generated:
2017_10_03-AM-10_48_48
Last ObjectModification:
2017_04_08-AM-11_48_43
Theory : reals
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