Nuprl Lemma : grp_eq_shift_right
∀[g:IGroup]. ∀[a,b:|g|]. uiff(a = b ∈ |g|;e = (b * (~ a)) ∈ |g|)
Proof
Definitions occuring in Statement :
igrp: IGroup,
grp_inv: ~,
grp_id: e,
grp_op: *,
grp_car: |g|,
uiff: uiff(P;Q),
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
igrp: IGroup,
imon: IMonoid,
infix_ap: x f y,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
grp_car_wf,
grp_id_wf,
grp_op_wf,
grp_inv_wf,
igrp_wf,
squash_wf,
true_wf,
mon_ident,
iff_weakening_equal,
infix_ap_wf,
grp_inverse,
uiff_wf,
mon_assoc,
iff_weakening_uiff,
grp_eq_op_r
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
because_Cache,
applyEquality,
independent_pairFormation,
addLevel,
independent_isectElimination,
lambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
cumulativity,
levelHypothesis
Latex:
\mforall{}[g:IGroup]. \mforall{}[a,b:|g|]. uiff(a = b;e = (b * (\msim{} a)))
Date html generated:
2017_10_01-AM-08_13_47
Last ObjectModification:
2017_02_28-PM-01_58_18
Theory : groups_1
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