Nuprl Lemma : int-to-ring-minus-one
∀[r:Rng]. (int-to-ring(r;-1) = (-r 1) ∈ |r|)
Proof
Definitions occuring in Statement :
int-to-ring: int-to-ring(r;n),
rng: Rng,
rng_one: 1,
rng_minus: -r,
rng_car: |r|,
uall: ∀[x:A]. B[x],
apply: f a,
minus: -n,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
int-to-ring: int-to-ring(r;n),
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
btrue: tt,
rng_nat_op: n ⋅r e,
mon_nat_op: n ⋅ e,
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
nat_op: n x(op;id) e,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
subtract: n - m,
uall: ∀[x:A]. B[x],
bfalse: ff,
member: t ∈ T,
squash: ↓T,
rng: Rng,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
infix_ap: x f y
Lemmas referenced :
equal_wf,
rng_car_wf,
rng_minus_over_plus,
rng_zero_wf,
rng_one_wf,
rng_minus_wf,
iff_weakening_equal,
rng_plus_wf,
rng_minus_zero,
rng_plus_zero,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
universeIsType
Latex:
\mforall{}[r:Rng]. (int-to-ring(r;-1) = (-r 1))
Date html generated:
2020_05_19-PM-10_07_58
Last ObjectModification:
2020_01_08-PM-06_00_23
Theory : rings_1
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