Nuprl Lemma : iabgrp_op_inv_assoc
∀[g:IAbGrp{i}]. ∀[a,b:|g|]. (((a * ((~ a) * b)) = b ∈ |g|) ∧ (((~ a) * (a * b)) = b ∈ |g|))
Proof
Definitions occuring in Statement :
iabgrp: IAbGrp{i},
grp_inv: ~,
grp_op: *,
grp_car: |g|,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
iabgrp: IAbGrp{i},
igrp: IGroup,
squash: ↓T,
imon: IMonoid,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
infix_ap: x f y,
prop: ℙ
Lemmas referenced :
mon_ident,
equal_wf,
mon_assoc,
grp_inv_wf,
grp_inv_assoc,
iff_weakening_equal,
grp_car_wf,
grp_op_wf,
grp_id_wf,
squash_wf,
true_wf,
iabgrp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
because_Cache,
addLevel,
applyEquality,
lambdaEquality,
imageElimination,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
independent_isectElimination,
independent_functionElimination,
productEquality,
universeEquality,
equalityUniverse,
levelHypothesis,
independent_pairEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[g:IAbGrp\{i\}]. \mforall{}[a,b:|g|]. (((a * ((\msim{} a) * b)) = b) \mwedge{} (((\msim{} a) * (a * b)) = b))
Date html generated:
2017_10_01-AM-08_13_51
Last ObjectModification:
2017_02_28-PM-01_58_20
Theory : groups_1
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