Nuprl Lemma : hgrp_of_ocgrp_wf2
∀[g:OGrp]. (g↓hgrp ∈ OCMon)
Proof
Definitions occuring in Statement :
hgrp_of_ocgrp: g↓hgrp,
ocgrp: OGrp,
ocmon: OCMon,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
hgrp_of_ocgrp: g↓hgrp,
grp_car: |g|,
pi1: fst(t),
grp_eq: =b,
pi2: snd(t),
grp_le: ≤b,
exists: ∃x:A. B[x],
ocgrp: OGrp,
hgrp_car: |g|+,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
so_apply: x[s1;s2],
infix_ap: x f y,
mon_hom_inj_p: IsMonHomInj(g;h;f),
inject: Inj(A;B;f),
monoid_hom_p: IsMonHom{M1,M2}(f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
grp_op: *,
grp_id: e,
cand: A c∧ B,
implies: P ⇒ Q,
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
inj_into_ocmon,
hgrp_of_ocgrp_wf,
ocgrp_wf,
hgrp_car_wf,
mon_hom_inj_p_wf,
rels_iso_wf,
grp_car_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_eq_wf,
subtype_rel_dep_function,
subtype_rel_self,
grp_le_wf,
exists_wf,
grp_op_wf,
grp_id_wf,
equal_wf,
grp_leq_wf,
hgrp_car_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
setElimination,
rename,
lambdaEquality,
independent_pairFormation,
productEquality,
because_Cache,
applyEquality,
functionEquality,
lambdaFormation,
isect_memberEquality,
productElimination,
dependent_set_memberEquality
Latex:
\mforall{}[g:OGrp]. (g\mdownarrow{}hgrp \mmember{} OCMon)
Date html generated:
2016_05_15-PM-00_14_21
Last ObjectModification:
2015_12_26-PM-11_41_19
Theory : groups_1
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