Nuprl Lemma : eqfun_p_shift
∀[A,B:Type]. ∀[eqa:A ⟶ A ⟶ 𝔹]. ∀[eqb:B ⟶ B ⟶ 𝔹]. ∀[f:A ⟶ B].
(IsEqFun(A;eqa)) supposing (IsEqFun(B;eqb) and RelsIso(A;B;x,y.↑(x eqa y);x,y.↑(x eqb y);f) and Inj(A;B;f))
Proof
Definitions occuring in Statement :
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
eqfun_p: IsEqFun(T;eq),
inject: Inj(A;B;f),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
eqfun_p: IsEqFun(T;eq),
rels_iso: RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f),
inject: Inj(A;B;f),
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
infix_ap: x f y,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
assert_wf,
assert_witness,
equal_wf,
eqfun_p_wf,
rels_iso_wf,
inject_wf,
bool_wf,
iff_transitivity,
iff_weakening_uiff,
uiff_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
applyEquality,
functionExtensionality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
lambdaEquality,
functionEquality,
universeEquality,
addLevel,
independent_pairFormation,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,B:Type]. \mforall{}[eqa:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}]. \mforall{}[eqb:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:A {}\mrightarrow{} B].
(IsEqFun(A;eqa)) supposing
(IsEqFun(B;eqb) and
RelsIso(A;B;x,y.\muparrow{}(x eqa y);x,y.\muparrow{}(x eqb y);f) and
Inj(A;B;f))
Date html generated:
2017_10_01-AM-08_13_10
Last ObjectModification:
2017_02_28-PM-01_57_28
Theory : gen_algebra_1
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