Nuprl Lemma : formal-sum-add-comm
∀[S:Type]. ∀[K:RngSig]. ∀[x,y:formal-sum(K;S)]. (x + y = y + x ∈ formal-sum(K;S))
Proof
Definitions occuring in Statement :
formal-sum-add: x + y,
formal-sum: formal-sum(K;S),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
rng_sig: RngSig
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
member: t ∈ T,
basic-formal-sum: basic-formal-sum(K;S),
formal-sum-add: x + y,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
quotient: x,y:A//B[x; y],
formal-sum: formal-sum(K;S),
all: ∀x:A. B[x],
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
rng_sig_wf,
formal-sum_wf,
iff_weakening_equal,
bag-append_wf,
bag-append-comm,
rng_car_wf,
bag_wf,
true_wf,
squash_wf,
equal_wf,
equal-wf-base,
formal-sum-add_wf1,
bfs-equiv_wf,
basic-formal-sum_wf,
quotient-member-eq,
bfs-equiv-rel,
formal-sum-add_functionality,
and_wf
Rules used in proof :
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
cumulativity,
productEquality,
because_Cache,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
introduction,
sqequalRule,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
pertypeElimination,
pointwiseFunctionalityForEquality,
dependent_functionElimination,
hyp_replacement,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[S:Type]. \mforall{}[K:RngSig]. \mforall{}[x,y:formal-sum(K;S)]. (x + y = y + x)
Date html generated:
2018_05_22-PM-09_45_31
Last ObjectModification:
2018_01_09-PM-01_00_15
Theory : linear!algebra
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