Nuprl Lemma : int_op_minus
∀[g:Group{i}]. ∀[e:|g|]. ∀[a:ℤ]. (-a x(*;e;~) e = (~ a x(*;e;~) e) ∈ |g|)
Proof
Definitions occuring in Statement :
int_op: i x(op;id;inv) e,
grp: Group{i},
grp_inv: ~,
grp_id: e,
grp_op: *,
grp_car: |g|,
uall: ∀[x:A]. B[x],
apply: f a,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
grp: Group{i},
mon: Mon,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
int_op: i x(op;id;inv) e,
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
imon: IMonoid,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
nat: ℕ,
le: A ≤ B
Lemmas referenced :
grp_car_wf,
grp_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
minus-zero,
equal_wf,
squash_wf,
true_wf,
nat_op_zero,
grp_sig_wf,
monoid_p_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
iff_weakening_equal,
grp_inv_id,
grp_subtype_igrp,
le_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
assert_wf,
le_wf,
eqtt_to_assert,
assert_of_le_int,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
intformnot_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
nat_op_wf,
grp_inv_inv,
nat_wf,
imon_wf,
minus-minus,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
intEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
extract_by_obid,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
setEquality,
imageMemberEquality,
baseClosed,
productElimination,
minusEquality,
lambdaFormation,
equalityElimination,
baseApply,
closedConclusion,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[g:Group\{i\}]. \mforall{}[e:|g|]. \mforall{}[a:\mBbbZ{}]. (-a x(*;e;\msim{}) e = (\msim{} a x(*;e;\msim{}) e))
Date html generated:
2017_10_01-AM-08_16_13
Last ObjectModification:
2017_02_28-PM-02_01_12
Theory : groups_1
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