Nuprl Lemma : any_field_is_integ_dom
∀[r:CRng]. IsIntegDom(r) supposing IsField(r)
Proof
Definitions occuring in Statement :
field_p: IsField(r),
integ_dom_p: IsIntegDom(r),
crng: CRng,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
integ_dom_p: IsIntegDom(r),
field_p: IsField(r),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
crng: CRng,
rng: Rng,
infix_ap: x f y,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
ring_divs: a | b in r,
exists: ∃x:A. B[x],
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
rng_car_wf,
rng_times_wf,
rng_zero_wf,
not_wf,
rng_one_wf,
nequal_wf,
all_wf,
ring_divs_wf,
crng_wf,
infix_ap_wf,
squash_wf,
true_wf,
rng_times_assoc,
rng_times_one,
iff_weakening_equal,
crng_times_comm,
rng_times_zero
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
applyEquality,
because_Cache,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
productEquality,
functionEquality,
isect_memberEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[r:CRng]. IsIntegDom(r) supposing IsField(r)
Date html generated:
2017_10_01-AM-08_17_38
Last ObjectModification:
2017_02_28-PM-02_02_54
Theory : rings_1
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