### Nuprl Lemma : rmul-nonneg

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

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]` or: `P ∨ Q` and: `P ∧ Q` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` rleq: `x ≤ y` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` or: `P ∨ Q` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  rmul_functionality rmul-assoc req_inversion rminus-zero rminus-rminus rmul-one-both rminus_functionality rmul_over_rminus req_transitivity rmul-minus rmul-int rless-int rmul_reverses_rleq_iff rminus_wf rmul_comm req_weakening rmul-zero-both rleq_functionality uiff_transitivity rmul_preserves_rleq2 rleq_wf and_wf or_wf nat_plus_wf real_wf int-to-real_wf rmul_wf rsub_wf less_than'_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality productElimination independent_pairEquality because_Cache lemma_by_obid isectElimination applyEquality hypothesis natural_numberEquality setElimination rename minusEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination unionElimination independent_isectElimination independent_functionElimination independent_pairFormation imageMemberEquality baseClosed multiplyEquality

Latex:
\mforall{}[x,y:\mBbbR{}].    r0  \mleq{}  (x  *  y)  supposing  ((r0  \mleq{}  x)  \mwedge{}  (r0  \mleq{}  y))  \mvee{}  ((x  \mleq{}  r0)  \mwedge{}  (y  \mleq{}  r0))

Date html generated: 2016_05_18-AM-07_33_32
Last ObjectModification: 2016_01_17-AM-02_01_20

Theory : reals

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