Nuprl Lemma : rmul_functionality_wrt_rleq

[x,y,z:ℝ].  ((x y) ≤ (z y)) supposing ((r0 ≤ y) and (x ≤ z))


Definitions occuring in Statement :  rleq: x ≤ y rmul: b int-to-real: r(n) real: uimplies: supposing a uall: [x:A]. B[x] natural_number: $n
Definitions unfolded in proof :  rleq: x ≤ y uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] implies:  Q rnonneg: rnonneg(x) le: A ≤ B and: P ∧ Q not: ¬A false: False subtype_rel: A ⊆B real: prop: itermConstant: "const" req_int_terms: t1 ≡ t2 top: Top uiff: uiff(P;Q) iff: ⇐⇒ Q rev_implies:  Q
Lemmas referenced :  rnonneg-rmul rsub_wf int-to-real_wf less_than'_wf rmul_wf real_wf nat_plus_wf rnonneg_wf real_term_polynomial itermSubtract_wf itermMultiply_wf itermVar_wf itermConstant_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_mul_lemma real_term_value_var_lemma req-iff-rsub-is-0 rnonneg_functionality
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination hypothesisEquality hypothesis natural_numberEquality independent_functionElimination lambdaEquality productElimination independent_pairEquality because_Cache applyEquality setElimination rename minusEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination computeAll int_eqEquality intEquality voidEquality independent_isectElimination

\mforall{}[x,y,z:\mBbbR{}].    ((x  *  y)  \mleq{}  (z  *  y))  supposing  ((r0  \mleq{}  y)  and  (x  \mleq{}  z))

Date html generated: 2017_10_03-AM-08_26_19
Last ObjectModification: 2017_07_28-AM-07_24_15

Theory : reals

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