Nuprl Lemma : rmin_lb

`∀[x,y,z:ℝ].  rmin(x;y) ≤ z supposing (x ≤ z) ∨ (y ≤ z)`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rmin: `rmin(x;y)` real: `ℝ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` or: `P ∨ Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` or: `P ∨ Q` and: `P ∧ Q` guard: `{T}` 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`
Lemmas referenced :  rmin-rleq rleq_transitivity rmin_wf less_than'_wf rsub_wf real_wf nat_plus_wf or_wf rleq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality unionElimination productElimination hypothesis independent_isectElimination sqequalRule lambdaEquality dependent_functionElimination independent_pairEquality because_Cache applyEquality setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination

Latex:
\mforall{}[x,y,z:\mBbbR{}].    rmin(x;y)  \mleq{}  z  supposing  (x  \mleq{}  z)  \mvee{}  (y  \mleq{}  z)

Date html generated: 2016_05_18-AM-07_16_55
Last ObjectModification: 2015_12_28-AM-00_44_03

Theory : reals

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