Nuprl Lemma : rless_functionality
∀x1,x2,y1,y2:ℝ. (x1 < y1 ⇐⇒ x2 < y2) supposing ((y1 = y2) and (x1 = x2))
Proof
Definitions occuring in Statement :
rless: x < y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
req: x = y,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
int_upper: {i...},
le: A ≤ B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
real: ℝ,
so_lambda: λ2x.t[x],
so_apply: x[s],
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
bfalse: ff
Lemmas referenced :
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
int_term_value_minus_lemma,
itermMinus_wf,
minus-is-int-iff,
not_wf,
bnot_wf,
assert_wf,
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,
decidable__lt,
lt_int_wf,
real_wf,
req_wf,
rless_wf,
all_wf,
int_upper_wf,
subtract_wf,
absval_ifthenelse,
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,
less_than_transitivity1,
rless-iff4,
less_than_wf,
rless-iff-large-diff,
req_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
independent_pairFormation,
dependent_functionElimination,
productElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
setElimination,
independent_isectElimination,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
applyEquality,
addEquality,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
imageElimination,
instantiate,
cumulativity,
impliesFunctionality
Latex:
\mforall{}x1,x2,y1,y2:\mBbbR{}. (x1 < y1 \mLeftarrow{}{}\mRightarrow{} x2 < y2) supposing ((y1 = y2) and (x1 = x2))
Date html generated:
2016_05_18-AM-07_04_57
Last ObjectModification:
2016_01_17-AM-01_51_25
Theory : reals
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