Nuprl Lemma : grp_inv_thru_op
∀[g:IGroup]. ∀[a,b:|g|]. ((~ (a * b)) = ((~ b) * (~ a)) ∈ |g|)
Proof
Definitions occuring in Statement :
igrp: IGroup,
grp_inv: ~,
grp_op: *,
grp_car: |g|,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
infix_ap: x f y,
igrp: IGroup,
imon: IMonoid,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
grp_op_cancel_l,
grp_op_wf,
grp_inv_wf,
grp_car_wf,
igrp_wf,
equal_wf,
squash_wf,
true_wf,
grp_inverse,
infix_ap_wf,
iff_weakening_equal,
grp_id_wf,
mon_assoc,
grp_inv_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[g:IGroup]. \mforall{}[a,b:|g|]. ((\msim{} (a * b)) = ((\msim{} b) * (\msim{} a)))
Date html generated:
2017_10_01-AM-08_13_43
Last ObjectModification:
2017_02_28-PM-01_58_09
Theory : groups_1
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