Nuprl Lemma : inverse_grp_sig_hom_shift
∀[g,h:GrpSig]. ∀[f:MonHom(g,h)].
(Inverse(|g|;*;e;~)) supposing (Inverse(|h|;*;e;~) and Inj(|g|;|h|;f) and fun_thru_1op(|g|;|h|;~;~;f))
Proof
Definitions occuring in Statement :
monoid_hom: MonHom(M1,M2),
grp_inv: ~,
grp_id: e,
grp_op: *,
grp_car: |g|,
grp_sig: GrpSig,
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
inverse: Inverse(T;op;id;inv),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
inverse: Inverse(T;op;id;inv),
inject: Inj(A;B;f),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
monoid_hom_p: IsMonHom{M1,M2}(f),
and: P ∧ Q,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
prop: ℙ,
monoid_hom: MonHom(M1,M2),
all: ∀x:A. B[x],
infix_ap: x f y,
implies: P ⇒ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
monoid_hom_properties,
grp_car_wf,
inverse_wf,
grp_op_wf,
grp_id_wf,
grp_inv_wf,
inject_wf,
fun_thru_1op_wf,
monoid_hom_wf,
grp_sig_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
independent_pairFormation,
dependent_functionElimination,
applyEquality,
independent_functionElimination,
lambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[g,h:GrpSig]. \mforall{}[f:MonHom(g,h)].
(Inverse(|g|;*;e;\msim{})) supposing
(Inverse(|h|;*;e;\msim{}) and
Inj(|g|;|h|;f) and
fun\_thru\_1op(|g|;|h|;\msim{};\msim{};f))
Date html generated:
2017_10_01-AM-08_14_11
Last ObjectModification:
2017_02_28-PM-01_58_36
Theory : groups_1
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