Nuprl Lemma : dot-product_functionality

[n:ℕ]. ∀[x1,y1,x2,y2:ℝ^n].  (x1 ⋅ y1 x2 ⋅ y2) supposing (req-vec(n;x1;x2) and req-vec(n;y1;y2))


Definitions occuring in Statement :  dot-product: x ⋅ y req-vec: req-vec(n;x;y) real-vec: ^n req: y nat: uimplies: supposing a uall: [x:A]. B[x]
Definitions unfolded in proof :  dot-product: x ⋅ y req-vec: req-vec(n;x;y) real-vec: ^n uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a nat: so_lambda: λ2x.t[x] int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q ge: i ≥  all: x:A. B[x] decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False implies:  Q not: ¬A top: Top prop: so_apply: x[s] uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  rmul_functionality rsum_functionality2 req_functionality req_weakening le_wf int_term_value_constant_lemma int_term_value_subtract_lemma int_formula_prop_le_lemma itermConstant_wf itermSubtract_wf intformle_wf nat_wf real_wf req_wf all_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 rmul_wf subtract_wf rsum_wf req_witness
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 independent_functionElimination equalityTransitivity equalitySymmetry functionEquality lambdaFormation

\mforall{}[n:\mBbbN{}].  \mforall{}[x1,y1,x2,y2:\mBbbR{}\^{}n].    (x1  \mcdot{}  y1  =  x2  \mcdot{}  y2)  supposing  (req-vec(n;x1;x2)  and  req-vec(n;y1;y2))

Date html generated: 2016_05_18-AM-09_47_11
Last ObjectModification: 2016_01_17-AM-02_51_07

Theory : reals

Home Index