Nuprl Lemma : rmul-rmax
∀[x,y,z:ℝ]. ((r0 ≤ z) ⇒ ((z * rmax(x;y)) = rmax(z * x;z * y)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmax: rmax(x;y),
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
rmax: rmax(x;y),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
reg-seq-mul: reg-seq-mul(x;y),
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
ge: i ≥ j ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
lt_int: i <z j,
subtract: n - m,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
absval: |i|
Lemmas referenced :
req-iff-bdd-diff,
rmul_wf,
rmax_wf,
rleq_wf,
int-to-real_wf,
req_witness,
real_wf,
reg-seq-mul_wf,
imax_wf,
nat_plus_wf,
bdd-diff_functionality,
rmul-bdd-diff-reg-seq-mul,
rmax_functionality_wrt_bdd-diff,
canonical-bound_wf,
add_nat_wf,
multiply_nat_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
subtype_rel_set,
int_upper_wf,
nat_wf,
le_wf,
absval_wf,
istype-int_upper,
upper_subtype_nat,
istype-false,
nat_properties,
add-is-int-iff,
multiply-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
false_wf,
subtract_wf,
divide_wfa,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
int_subtype_base,
nequal_wf,
decidable__equal_int,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
squash_wf,
true_wf,
int_nzero_wf,
imax_unfold,
istype-nat,
subtype_rel_self,
iff_weakening_equal,
absval_ifthenelse,
minus-one-mul,
mul-commutes,
add-mul-special,
zero-mul,
neg-approx-of-nonneg-real,
mul_preserves_le,
decidable__lt,
istype-less_than,
div_preserves_le,
le_functionality,
int-triangle-inequality2,
le_weakening,
sq_stable__le,
int_upper_properties,
absval_div_nat,
absval_mul,
div-cancel,
multiply_functionality_wrt_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
universeIsType,
natural_numberEquality,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
applyEquality,
setElimination,
rename,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
addEquality,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
approximateComputation,
voidElimination,
functionEquality,
independent_pairFormation,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
equalityIstype,
functionIsType,
sqequalBase,
intEquality,
equalityElimination,
instantiate,
imageElimination,
imageMemberEquality,
universeEquality,
minusEquality,
cumulativity
Latex:
\mforall{}[x,y,z:\mBbbR{}]. ((r0 \mleq{} z) {}\mRightarrow{} ((z * rmax(x;y)) = rmax(z * x;z * y)))
Date html generated:
2019_10_29-AM-10_03_54
Last ObjectModification:
2019_04_01-PM-11_11_22
Theory : reals
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