Nuprl Lemma : rless_transitivity
∀x,y,z:ℝ. ((x < y) ⇒ (y < z) ⇒ (x < z))
Proof
Definitions occuring in Statement :
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
or: P ∨ Q,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
real: ℝ,
int_upper: {i...},
le: A ≤ B,
guard: {T},
so_apply: x[s],
uiff: uiff(P;Q),
cand: A c∧ B,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T
Lemmas referenced :
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
le_wf,
imax_ub,
int_upper_subtype_int_upper,
imax_lb,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_upper_properties,
less_than_transitivity1,
all_wf,
int_upper_wf,
imax_wf,
less_than_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
imax_strict_ub,
real_wf,
rless_wf,
rless-iff4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
isectElimination,
natural_numberEquality,
setElimination,
rename,
inlFormation,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
addEquality,
applyEquality,
because_Cache,
inrFormation,
setEquality,
imageElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}x,y,z:\mBbbR{}. ((x < y) {}\mRightarrow{} (y < z) {}\mRightarrow{} (x < z))
Date html generated:
2016_05_18-AM-07_05_34
Last ObjectModification:
2016_01_17-AM-01_52_22
Theory : reals
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