Nuprl Lemma : rless*_transitivity2
∀x,y,z:ℝ*. (x ≤ y ⇒ y < z ⇒ x < z)
Proof
Definitions occuring in Statement :
rleq*: x ≤ y,
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,
rleq*: x ≤ y,
rrel*: R*(x,y),
exists: ∃x:A. B[x],
rless*: x < y,
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
rless: x < y,
sq_exists: ∃x:A [B[x]],
real*: ℝ*,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
imax_wf,
imax_nat,
nat_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
int_upper_subtype_int_upper,
imax_ub,
int_upper_properties,
sq_stable__less_than,
int_upper_subtype_nat,
real_wf,
nat_plus_properties,
rless_transitivity2,
int_upper_wf,
all_wf,
rless_wf,
rless*_wf,
rleq*_wf,
real*_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
sqequalRule,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyEquality,
because_Cache,
inrFormation,
inlFormation,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}x,y,z:\mBbbR{}*. (x \mleq{} y {}\mRightarrow{} y < z {}\mRightarrow{} x < z)
Date html generated:
2018_05_22-PM-03_20_09
Last ObjectModification:
2017_10_06-PM-04_25_44
Theory : reals_2
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