### Nuprl Lemma : dot-product-nonneg

`∀[n:ℕ]. ∀[x:ℝ^n].  (r0 ≤ x ⋅ x)`

Proof

Definitions occuring in Statement :  dot-product: `x ⋅ y` real-vec: `ℝ^n` rleq: `x ≤ y` int-to-real: `r(n)` nat: `ℕ` uall: `∀[x:A]. B[x]` natural_number: `\$n`
Definitions unfolded in proof :  dot-product: `x ⋅ y` real-vec: `ℝ^n` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` 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` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` less_than: `a < b` so_apply: `x[s]` pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]` rleq: `x ≤ y` rnonneg: `rnonneg(x)` nat_plus: `ℕ+` subtype_rel: `A ⊆r B`
Lemmas referenced :  nat_wf real_wf nat_plus_wf int-to-real_wf nat_plus_properties rsum_wf rsub_wf less_than'_wf le_wf int_term_value_constant_lemma int_term_value_subtract_lemma int_formula_prop_le_lemma itermConstant_wf itermSubtract_wf intformle_wf square-nonneg 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 rmul_wf subtract_wf rsum_nonneg
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis lambdaEquality applyEquality dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache addEquality lambdaFormation independent_pairEquality minusEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[x:\mBbbR{}\^{}n].    (r0  \mleq{}  x  \mcdot{}  x)

Date html generated: 2016_05_18-AM-09_47_56
Last ObjectModification: 2016_01_17-AM-02_52_15

Theory : reals

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