Nuprl Lemma : rng_hom_minus
∀[r,s:Rng]. ∀[f:|r| ⟶ |s|]. ∀[x:|r|]. (f[-r x] = (-s f[x]) ∈ |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_minus: -r,
rng_car: |r|,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rng_hom_p: rng_hom_p(r;s;f),
and: P ∧ Q,
rng: Rng,
prop: ℙ,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
so_apply: x[s],
infix_ap: x f y,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_hom_zero,
rng_car_wf,
rng_hom_p_wf,
rng_wf,
rng_minus_wf,
infix_ap_wf,
rng_plus_wf,
equal_wf,
squash_wf,
true_wf,
rng_plus_comm,
rng_plus_inv,
iff_weakening_equal,
rng_plus_ac_1,
rng_plus_zero
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
productElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
setElimination,
rename,
functionExtensionality,
applyEquality,
because_Cache,
functionEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[r,s:Rng]. \mforall{}[f:|r| {}\mrightarrow{} |s|]. \mforall{}[x:|r|]. (f[-r x] = (-s f[x])) supposing rng\_hom\_p(r;s;f)
Date html generated:
2017_10_01-AM-08_18_23
Last ObjectModification:
2017_02_28-PM-02_03_17
Theory : rings_1
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