### Nuprl Lemma : rmin-nonneg

`∀[x,y:ℝ].  rnonneg(rmin(x;y)) supposing rnonneg(x) ∧ rnonneg(y)`

Proof

Definitions occuring in Statement :  rnonneg: `rnonneg(x)` rmin: `rmin(x;y)` real: `ℝ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` and: `P ∧ Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` rmin: `rmin(x;y)` squash: `↓T` prop: `ℙ` le: `A ≤ B` real: `ℝ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A`
Lemmas referenced :  le_wf squash_wf true_wf imin_unfold iff_weakening_equal le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot nat_plus_wf less_than'_wf rmin_wf real_wf rnonneg_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin lambdaFormation hypothesis dependent_functionElimination hypothesisEquality sqequalRule applyEquality lambdaEquality imageElimination extract_by_obid isectElimination equalityTransitivity equalitySymmetry intEquality setElimination rename because_Cache natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination independent_functionElimination unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate cumulativity voidElimination independent_pairEquality minusEquality axiomEquality productEquality isect_memberEquality

Latex:
\mforall{}[x,y:\mBbbR{}].    rnonneg(rmin(x;y))  supposing  rnonneg(x)  \mwedge{}  rnonneg(y)

Date html generated: 2017_10_03-AM-08_24_39
Last ObjectModification: 2017_07_28-AM-07_23_26

Theory : reals

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