Nuprl Lemma : sq_stable__mprime
∀g:GrpSig. ((∀a,b:|g|. Dec(a | b)) ⇒ (∀c:|g|. SqStable(IsPrime(c))))
Proof
Definitions occuring in Statement :
mprime: IsPrime(a),
mdivides: b | a,
sq_stable: SqStable(P),
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
grp_car: |g|,
grp_sig: GrpSig
Definitions unfolded in proof :
mprime: IsPrime(a),
sq_stable: SqStable(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
squash: ↓T,
false: False,
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
decidable__or,
sq_stable_from_decidable,
grp_sig_wf,
decidable_wf,
or_wf,
all_wf,
not_wf,
and_wf,
squash_wf,
grp_car_wf,
grp_op_wf,
mdivides_wf,
munit_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
independent_functionElimination,
hypothesis,
voidElimination,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_pairFormation,
applyEquality,
isectElimination,
lambdaEquality,
functionEquality,
because_Cache,
introduction,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:GrpSig. ((\mforall{}a,b:|g|. Dec(a | b)) {}\mRightarrow{} (\mforall{}c:|g|. SqStable(IsPrime(c))))
Date html generated:
2016_05_16-AM-07_43_57
Last ObjectModification:
2016_01_16-PM-11_38_04
Theory : factor_1
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