### Nuprl Lemma : rmul_reverses_rleq_iff

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

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rless: `x < y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` uiff: `uiff(P;Q)` and: `P ∧ Q` rleq: `x ≤ y` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` guard: `{T}` rneq: `x ≠ y` or: `P ∨ Q` rdiv: `(x/y)` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` label: `...\$L... t` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` top: `Top`
Lemmas referenced :  less_than'_wf rsub_wf rmul_wf real_wf nat_plus_wf rleq_wf rless_wf int-to-real_wf rmul_reverses_rleq rdiv_wf rinv-negative rleq_functionality_wrt_implies rinv_wf2 rleq_weakening_rless rless-implies-rless real_term_polynomial itermSubtract_wf itermConstant_wf itermVar_wf itermMultiply_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_var_lemma real_term_value_mul_lemma req-iff-rsub-is-0 rleq_weakening_equal rleq_weakening req_wf req_weakening uiff_transitivity rleq_functionality req_functionality req_inversion rmul-assoc rmul_functionality rmul_comm req_transitivity rmul-ac rmul-rdiv-cancel rmul-one-both
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality productElimination independent_pairEquality because_Cache extract_by_obid isectElimination applyEquality hypothesis setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry voidElimination isect_memberEquality independent_isectElimination lemma_by_obid inlFormation independent_functionElimination computeAll int_eqEquality intEquality voidEquality

Latex:
\mforall{}[x,y,z:\mBbbR{}].    uiff(x  \mleq{}  z;(z  *  y)  \mleq{}  (x  *  y))  supposing  y  <  r0

Date html generated: 2017_10_03-AM-08_35_00
Last ObjectModification: 2017_07_28-AM-07_28_45

Theory : reals

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