Nuprl Lemma : zero_ideal_wf
∀[r:CRng]. ((0r) ∈ Ideal(r){i})
Proof
Definitions occuring in Statement :
zero_ideal: (0r),
ideal: Ideal(r){i},
crng: CRng,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
zero_ideal: (0r),
ideal: Ideal(r){i},
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng,
ideal_p: S Ideal of R,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subgrp_p: s SubGrp of g,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
grp_car: |g|,
grp_inv: ~,
grp_op: *,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
crng_wf,
equal_wf,
rng_car_wf,
rng_zero_wf,
squash_wf,
true_wf,
rng_minus_zero,
iff_weakening_equal,
rng_minus_wf,
rng_plus_zero,
rng_plus_wf,
member_wf,
rng_times_zero,
rng_times_wf,
ideal_p_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
dependent_set_memberEquality,
lambdaEquality,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
because_Cache,
independent_pairFormation,
lambdaFormation,
applyEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
instantiate,
functionExtensionality
Latex:
\mforall{}[r:CRng]. ((0r) \mmember{} Ideal(r)\{i\})
Date html generated:
2017_10_01-AM-08_17_42
Last ObjectModification:
2017_02_28-PM-02_03_01
Theory : rings_1
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