Nuprl Lemma : inj_into_ocmon
∀[g:GrpSig]
g ∈ OCMon
supposing ∃h:OCMon
∃f:|g| ⟶ |h|
(IsMonHomInj(g;h;f)
∧ RelsIso(|g|;|h|;x,y.↑(x =b y);x,y.↑(x =b y);f)
∧ RelsIso(|g|;|h|;x,y.↑(x ≤b y);x,y.↑(x ≤b y);f))
Proof
Definitions occuring in Statement :
ocmon: OCMon,
mon_hom_inj_p: IsMonHomInj(g;h;f),
grp_le: ≤b,
grp_eq: =b,
grp_car: |g|,
grp_sig: GrpSig,
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
mon_hom_inj_p: IsMonHomInj(g;h;f),
ulinorder: UniformLinorder(T;x,y.R[x; y]),
monoid_p: IsMonoid(T;op;id),
monoid_hom_p: IsMonHom{M1,M2}(f),
uorder: UniformOrder(T;x,y.R[x; y]),
monoid_hom: MonHom(M1,M2),
cand: A c∧ B,
monot: monot(T;x,y.R[x; y];f),
implies: P ⇒ Q,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
urefl: UniformlyRefl(T;x,y.E[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f)
Lemmas referenced :
ocmon_wf,
grp_car_wf,
mon_hom_inj_p_wf,
rels_iso_wf,
assert_wf,
grp_eq_wf,
grp_le_wf,
grp_sig_wf,
ocmon_properties,
abmonoid_properties,
mon_properties,
comm_wf,
grp_op_wf,
monoid_p_wf,
grp_id_wf,
assoc_shift,
ident_mon_hom_shift,
monoid_hom_p_wf,
comm_shift,
assert_witness,
infix_ap_wf,
bool_wf,
istype-assert,
urefl_shift,
utrans_shift,
uanti_sym_shift,
connex_shift,
iff_imp_equal_bool,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bfalse_wf,
iff_weakening_uiff,
equal_wf,
assert_of_mon_eq,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
abdmonoid_wf,
dmon_wf,
assert_of_band,
cancel_shift,
monot_shift,
ulinorder_wf,
cancel_wf,
monot_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productIsType,
universeIsType,
extract_by_obid,
functionIsType,
isectElimination,
hypothesisEquality,
setElimination,
rename,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
because_Cache,
isect_memberEquality_alt,
isectIsTypeImplies,
dependent_functionElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
independent_isectElimination,
independent_functionElimination,
functionIsTypeImplies,
functionExtensionality_alt,
unionElimination,
instantiate,
lambdaFormation_alt,
promote_hyp,
equalityIstype,
cumulativity,
productEquality,
hyp_replacement,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
isectIsType
Latex:
\mforall{}[g:GrpSig]
g \mmember{} OCMon
supposing \mexists{}h:OCMon
\mexists{}f:|g| {}\mrightarrow{} |h|
(IsMonHomInj(g;h;f)
\mwedge{} RelsIso(|g|;|h|;x,y.\muparrow{}(x =\msubb{} y);x,y.\muparrow{}(x =\msubb{} y);f)
\mwedge{} RelsIso(|g|;|h|;x,y.\muparrow{}(x \mleq{}\msubb{} y);x,y.\muparrow{}(x \mleq{}\msubb{} y);f))
Date html generated:
2019_10_15-AM-10_32_50
Last ObjectModification:
2018_11_27-AM-10_30_58
Theory : groups_1
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