### Nuprl Lemma : rmin_strict_ub

`∀x,y,z:ℝ.  ((z < x) ∧ (z < y) `⇐⇒` z < rmin(x;y))`

Proof

Definitions occuring in Statement :  rless: `x < y` rmin: `rmin(x;y)` real: `ℝ` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` nat_plus: `ℕ+` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` rmin: `rmin(x;y)` squash: `↓T` real: `ℝ` int_upper: `{i...}` le: `A ≤ B` true: `True` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  rless_wf rmin_wf real_wf rless-iff4 imax_nat_plus nat_plus_wf nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf imax_wf less_than_wf squash_wf true_wf less_than_transitivity1 imin_unfold iff_weakening_equal int_upper_wf all_wf int_upper_properties decidable__le intformle_wf int_formula_prop_le_lemma imax_lb le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot le_wf int_upper_subtype_int_upper imax_ub rmin-rleq rless_transitivity1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin productEquality cut introduction extract_by_obid isectElimination hypothesisEquality hypothesis dependent_functionElimination independent_functionElimination because_Cache equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename unionElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule computeAll dependent_set_memberEquality applyEquality imageElimination addEquality imageMemberEquality baseClosed universeEquality equalityElimination promote_hyp instantiate cumulativity inrFormation inlFormation

Latex:
\mforall{}x,y,z:\mBbbR{}.    ((z  <  x)  \mwedge{}  (z  <  y)  \mLeftarrow{}{}\mRightarrow{}  z  <  rmin(x;y))

Date html generated: 2017_10_03-AM-08_30_20
Last ObjectModification: 2017_07_28-AM-07_26_31

Theory : reals

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