Nuprl Lemma : rng_hom_zero
∀[r,s:Rng]. ∀[f:|r| ⟶ |s|]. f[0] = 0 ∈ |s| supposing rng_hom_p(r;s;f)
Proof
Definitions occuring in Statement :
rng_hom_p: rng_hom_p(r;s;f),
rng: Rng,
rng_zero: 0,
rng_car: |r|,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
rng_hom_p: rng_hom_p(r;s;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
rng: Rng,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
infix_ap: x f y,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
rng_plus_zero,
rng_zero_wf,
iff_weakening_equal,
fun_thru_2op_wf,
rng_car_wf,
rng_plus_wf,
rng_times_wf,
rng_one_wf,
rng_wf,
rng_minus_wf,
infix_ap_wf,
rng_plus_comm,
rng_plus_ac_1,
rng_plus_inv
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
productEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[r,s:Rng]. \mforall{}[f:|r| {}\mrightarrow{} |s|]. f[0] = 0 supposing rng\_hom\_p(r;s;f)
Date html generated:
2017_10_01-AM-08_18_19
Last ObjectModification:
2017_02_28-PM-02_03_14
Theory : rings_1
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