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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