Nuprl Lemma : eqv_mod_subset_is_eqv
∀g:IGroup
∀[s:|g| ⟶ ℙ]
((s e)
⇒ (∀a:|g|. ((s a) ⇒ (s (~ a))))
⇒ (∀a,b:|g|. ((s a) ⇒ (s b) ⇒ (s (a * b))))
⇒ EquivRel(|g|;x,y.x ≡ y (mod s in g)))
Proof
Definitions occuring in Statement :
eqv_mod_subset: a ≡ b (mod s in g),
igrp: IGroup,
grp_inv: ~,
grp_id: e,
grp_op: *,
grp_car: |g|,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
igrp: IGroup,
imon: IMonoid,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
eqv_mod_subset: a ≡ b (mod s in g),
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
cand: A c∧ B,
infix_ap: x f y,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
grp_car_wf,
infix_ap_wf,
grp_op_wf,
grp_inv_wf,
grp_id_wf,
igrp_wf,
grp_inverse,
iff_weakening_equal,
grp_inv_thru_op,
grp_inv_inv,
mon_assoc,
grp_inv_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
applyEquality,
universeEquality,
cumulativity,
independent_pairFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}g:IGroup
\mforall{}[s:|g| {}\mrightarrow{} \mBbbP{}]
((s e)
{}\mRightarrow{} (\mforall{}a:|g|. ((s a) {}\mRightarrow{} (s (\msim{} a))))
{}\mRightarrow{} (\mforall{}a,b:|g|. ((s a) {}\mRightarrow{} (s b) {}\mRightarrow{} (s (a * b))))
{}\mRightarrow{} EquivRel(|g|;x,y.x \mequiv{} y (mod s in g)))
Date html generated:
2016_05_15-PM-00_09_07
Last ObjectModification:
2015_12_26-PM-11_46_12
Theory : groups_1
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