### 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))`

Proof

Definitions occuring in Statement :  dot-product: `x ⋅ y` req-vec: `req-vec(n;x;y)` real-vec: `ℝ^n` req: `x = y` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]`
Definitions unfolded in proof :  dot-product: `x ⋅ y` req-vec: `req-vec(n;x;y)` real-vec: `ℝ^n` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` 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` 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]` 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

Latex:
\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

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