Nuprl Lemma : rless_transitivity1

`∀x,y,z:ℝ.  ((x < y) `` x < z supposing y ≤ z)`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rless: `x < y` real: `ℝ` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` and: `P ∧ Q` not: `¬A` false: `False` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` iff: `P `⇐⇒` Q` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` rev_implies: `P `` Q` exists: `∃x:A. B[x]` int_upper: `{i...}` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` uiff: `uiff(P;Q)`
Lemmas referenced :  false_wf int_term_value_subtract_lemma int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_less_lemma itermSubtract_wf itermConstant_wf itermAdd_wf intformless_wf subtract-is-int-iff add-is-int-iff decidable__lt int_formula_prop_wf int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_plus_properties int_upper_properties rless_wf rleq_wf all_wf int_upper_wf less_than_transitivity1 rless-iff4 rleq-iff4 less_than_wf rless-iff-large-diff nat_plus_wf real_wf rsub_wf less_than'_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality productElimination independent_pairEquality voidElimination lemma_by_obid isectElimination applyEquality hypothesis setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry independent_functionElimination dependent_set_memberEquality independent_pairFormation imageMemberEquality baseClosed dependent_pairFormation independent_isectElimination addEquality because_Cache unionElimination int_eqEquality intEquality isect_memberEquality voidEquality computeAll pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}x,y,z:\mBbbR{}.    ((x  <  y)  {}\mRightarrow{}  x  <  z  supposing  y  \mleq{}  z)

Date html generated: 2016_05_18-AM-07_05_39
Last ObjectModification: 2016_01_17-AM-01_50_59

Theory : reals

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