Nuprl Lemma : princ_ideal_wf
∀[r:Rng]. ∀[a:|r|]. ((a)r ∈ Ideal(r){i})
Proof
Definitions occuring in Statement :
princ_ideal: (a)r,
ideal: Ideal(r){i},
rng: Rng,
rng_car: |r|,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng: Rng,
ideal: Ideal(r){i},
so_apply: x[s],
infix_ap: x f y,
so_lambda: λ2x.t[x],
princ_ideal: (a)r,
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
grp_op: *,
grp_inv: ~,
grp_car: |g|,
pi1: fst(t),
pi2: snd(t),
grp_id: e,
add_grp_of_rng: r↓+gp,
subgrp_p: s SubGrp of g,
ideal_p: S Ideal of R,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T
Lemmas referenced :
rng_car_wf,
rng_wf,
rng_times_wf,
equal_wf,
exists_wf,
rng_times_zero,
rng_zero_wf,
rng_times_over_minus,
iff_weakening_equal,
infix_ap_wf,
true_wf,
squash_wf,
rng_minus_wf,
rng_times_over_plus,
rng_plus_wf,
rng_times_assoc,
subgrp_p_wf,
add_grp_of_rng_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality_alt,
because_Cache,
dependent_set_memberEquality_alt,
applyEquality,
lambdaEquality,
productElimination,
lambdaFormation,
independent_pairFormation,
dependent_pairFormation,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
imageElimination,
cumulativity,
functionEquality,
productIsType,
functionIsType,
inhabitedIsType,
instantiate
Latex:
\mforall{}[r:Rng]. \mforall{}[a:|r|]. ((a)r \mmember{} Ideal(r)\{i\})
Date html generated:
2019_10_15-AM-10_33_28
Last ObjectModification:
2018_10_08-AM-09_08_21
Theory : rings_1
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