Nuprl Lemma : int_add_grp_wf2
<ℤ+> ∈ OGrp
Proof
Definitions occuring in Statement :
int_add_grp: <ℤ+>,
ocgrp: OGrp,
member: t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
ocgrp: OGrp,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
abgrp: AbGrp,
grp: Group{i},
int_add_grp: <ℤ+>,
grp_car: |g|,
pi1: fst(t),
grp_le: ≤b,
pi2: snd(t),
grp_eq: =b,
grp_op: *,
le: A ≤ B,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
monot: monot(T;x,y.R[x; y];f),
ulinorder: UniformLinorder(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
utrans: UniformlyTrans(T;x,y.E[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
cancel: Cancel(T;S;op),
inverse: Inverse(T;op;id;inv),
grp_inv: ~,
grp_id: e,
cand: A c∧ B
Lemmas referenced :
inverse_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
grp_inv_wf,
ulinorder_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
uall_wf,
monot_wf,
int_add_grp_wf,
subtype_rel_sets,
mon_wf,
comm_wf,
set_wf,
ulinorder_functionality_wrt_iff,
le_int_wf,
le_wf,
assert_of_le_int,
less_than'_wf,
assert_witness,
monot_functionality,
iff_weakening_uiff,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformand_wf,
int_formula_prop_and_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__or,
intformor_wf,
int_formula_prop_or_lemma,
iff_imp_equal_bool,
eq_int_wf,
band_wf,
equal-wf-base,
int_subtype_base,
assert_of_eq_int,
iff_transitivity,
assert_of_band,
iff_wf,
itermAdd_wf,
int_term_value_add_lemma,
itermMinus_wf,
itermConstant_wf,
int_term_value_minus_lemma,
int_term_value_constant_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
productEquality,
sqequalRule,
lambdaEquality,
functionEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
instantiate,
setEquality,
cumulativity,
independent_pairFormation,
intEquality,
isect_memberFormation,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
voidElimination,
addEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
functionExtensionality,
addLevel,
impliesFunctionality,
baseApply,
closedConclusion,
baseClosed
Latex:
<\mBbbZ{}+> \mmember{} OGrp
Date html generated:
2017_10_01-AM-08_16_56
Last ObjectModification:
2017_02_28-PM-02_02_34
Theory : groups_1
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