Nuprl Lemma : mon_itop_split_el
∀[g:IMonoid]. ∀[a,b,c:ℤ].
(∀[E:{a..c-} ⟶ |g|]
((Π a ≤ j < c. E[j]) = ((Π a ≤ j < b. E[j]) * (E[b] * (Π b + 1 ≤ j < c. E[j]))) ∈ |g|)) supposing
(b < c and
(a ≤ b))
Proof
Definitions occuring in Statement :
mon_itop: Π lb ≤ i < ub. E[i],
imon: IMonoid,
grp_op: *,
grp_car: |g|,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
imon: IMonoid,
prop: ℙ,
squash: ↓T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
int_seg_wf,
grp_car_wf,
less_than_wf,
le_wf,
imon_wf,
equal_wf,
squash_wf,
true_wf,
mon_itop_split,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
infix_ap_wf,
grp_op_wf,
mon_itop_wf,
decidable__lt,
lelt_wf,
itermAdd_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
iff_weakening_equal,
mon_itop_unroll_lo
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
intEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
addEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[g:IMonoid]. \mforall{}[a,b,c:\mBbbZ{}].
(\mforall{}[E:\{a..c\msupminus{}\} {}\mrightarrow{} |g|]
((\mPi{} a \mleq{} j < c. E[j]) = ((\mPi{} a \mleq{} j < b. E[j]) * (E[b] * (\mPi{} b + 1 \mleq{} j < c. E[j]))))) supposing
(b < c and
(a \mleq{} b))
Date html generated:
2017_10_01-AM-08_16_17
Last ObjectModification:
2017_02_28-PM-02_01_17
Theory : groups_1
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