Nuprl Lemma : ball_functionality_wrt_permr
∀T:Type. ∀as,bs:T List. ∀P,Q:T ⟶ 𝔹.  ((as ≡(T) bs) 
⇒ (∀x:T. P[x] = Q[x]) 
⇒ ∀bx(:T) ∈ as. P[x] = ∀bx(:T) ∈ bs. Q[x])
Proof
Definitions occuring in Statement : 
ball: ball, 
permr: as ≡(T) bs
, 
list: T List
, 
bool: 𝔹
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
ball: ball, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
so_apply: x[s]
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
true: True
, 
squash: ↓T
, 
bool: 𝔹
, 
grp_car: |g|
, 
pi1: fst(t)
, 
band_mon: <𝔹,∧b>
, 
guard: {T}
, 
uimplies: b supposing a
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
istype-universe, 
bool_wf, 
permr_wf, 
list_wf, 
band_mon_wf, 
abmonoid_subtype_iabmonoid, 
mem_f_wf, 
mon_for_wf, 
equal_wf, 
squash_wf, 
true_wf, 
mon_for_functionality_wrt_permr, 
subtype_rel_self, 
grp_car_wf, 
mon_subtype_grp_sig, 
abmonoid_subtype_mon, 
subtype_rel_transitivity, 
abmonoid_wf, 
mon_wf, 
grp_sig_wf, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
cut, 
hypothesis, 
sqequalRule, 
functionIsType, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
equalityIsType1, 
universeIsType, 
applyEquality, 
dependent_functionElimination, 
inhabitedIsType, 
universeEquality, 
lambdaEquality_alt, 
because_Cache, 
natural_numberEquality, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
instantiate, 
independent_isectElimination, 
imageMemberEquality, 
baseClosed, 
productElimination
Latex:
\mforall{}T:Type.  \mforall{}as,bs:T  List.  \mforall{}P,Q:T  {}\mrightarrow{}  \mBbbB{}.
    ((as  \mequiv{}(T)  bs)  {}\mRightarrow{}  (\mforall{}x:T.  P[x]  =  Q[x])  {}\mRightarrow{}  \mforall{}\msubb{}x(:T)  \mmember{}  as.  P[x]  =  \mforall{}\msubb{}x(:T)  \mmember{}  bs.  Q[x])
Date html generated:
2019_10_16-PM-01_03_17
Last ObjectModification:
2018_10_08-AM-11_25_53
Theory : list_2
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