Nuprl Lemma : ideal_defines_eqv
∀r:CRng. ∀[a:|r| ⟶ ℙ]. (a Ideal of r ⇒ EquivRel(|r|;u,v.a (u +r (-r v))))
Proof
Definitions occuring in Statement :
ideal_p: S Ideal of R,
crng: CRng,
rng_minus: -r,
rng_plus: +r,
rng_car: |r|,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
ideal_p: S Ideal of R,
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
member: t ∈ T,
crng: CRng,
rng: Rng,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subgrp_p: s SubGrp of g,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
sym: Sym(T;x,y.E[x; y]),
infix_ap: x f y,
trans: Trans(T;x,y.E[x; y]),
grp_car: |g|,
grp_op: *
Lemmas referenced :
rng_car_wf,
ideal_p_wf,
crng_wf,
rng_plus_inv,
iff_weakening_equal,
rng_plus_wf,
rng_minus_wf,
rng_one_wf,
rng_plus_comm,
rng_times_over_plus,
rng_times_over_minus,
rng_times_one,
rng_minus_minus,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_inv_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
applyEquality,
lambdaEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
because_Cache
Latex:
\mforall{}r:CRng. \mforall{}[a:|r| {}\mrightarrow{} \mBbbP{}]. (a Ideal of r {}\mRightarrow{} EquivRel(|r|;u,v.a (u +r (-r v))))
Date html generated:
2016_05_15-PM-00_23_17
Last ObjectModification:
2015_12_27-AM-00_01_00
Theory : rings_1
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