Nuprl Lemma : grp_hom_inv
∀[g,h:IGroup]. ∀[f:MonHom(g,h)]. ∀[a:|g|]. ((f (~ a)) = (~ (f a)) ∈ |h|)
Proof
Definitions occuring in Statement :
monoid_hom: MonHom(M1,M2),
igrp: IGroup,
grp_inv: ~,
grp_car: |g|,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
igrp: IGroup,
imon: IMonoid,
monoid_hom: MonHom(M1,M2),
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
grp_car_wf,
monoid_hom_wf,
igrp_wf,
grp_eq_op_l,
grp_inv_wf,
equal_wf,
squash_wf,
true_wf,
monoid_hom_op,
infix_ap_wf,
grp_op_wf,
iff_weakening_equal,
grp_inverse,
monoid_hom_id,
grp_id_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
productElimination,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[g,h:IGroup]. \mforall{}[f:MonHom(g,h)]. \mforall{}[a:|g|]. ((f (\msim{} a)) = (\msim{} (f a)))
Date html generated:
2017_10_01-AM-08_14_03
Last ObjectModification:
2017_02_28-PM-01_58_14
Theory : groups_1
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