Nuprl Lemma : t-sqle-apply
∀[A,B:Type]. ∀a1,a2:A. ∀f1,f2:a:A ⟶ B. (t-sqle(a:A ⟶ B;f1;f2) ⇒ t-sqle(A;a1;a2) ⇒ t-sqle(B;f1 a1;f2 a2))
Proof
Definitions occuring in Statement :
t-sqle: t-sqle(T;a;b),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
t-sqle: t-sqle(T;a;b),
squash: ↓T,
exists: ∃x:A. B[x],
per-class: per-class(T;a),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
t-sqle_wf,
base_wf,
exists_wf,
sqle_wf_base,
is-exception_wf,
has-value_wf_base,
equal-wf-base-T
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
setElimination,
rename,
dependent_pairFormation,
applyEquality,
equalitySymmetry,
hypothesis,
dependent_set_memberEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
lemma_by_obid,
isectElimination,
equalityTransitivity,
divergentSqle,
sqleRule,
setEquality,
lambdaEquality,
because_Cache,
imageMemberEquality,
functionEquality,
dependent_functionElimination,
universeEquality,
isect_memberEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}a1,a2:A. \mforall{}f1,f2:a:A {}\mrightarrow{} B. (t-sqle(a:A {}\mrightarrow{} B;f1;f2) {}\mRightarrow{} t-sqle(A;a1;a2) {}\mRightarrow{} t-sqle(B;f1 a1;f2 a2))
Date html generated:
2016_05_13-PM-04_12_54
Last ObjectModification:
2016_01_14-PM-07_28_48
Theory : subtype_1
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