### Nuprl Lemma : rmul_functionality_wrt_rleq

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

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` uimplies: `b 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: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` rnonneg: `rnonneg(x)` le: `A ≤ B` and: `P ∧ Q` not: `¬A` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` top: `Top` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` 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

Latex:
\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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