Nuprl Lemma : equipollent-rationals
ℚ ~ {p:ℤ × ℕ+| ↑is-qrep(p)}
Proof
Definitions occuring in Statement :
is-qrep: is-qrep(p),
rationals: ℚ,
equipollent: A ~ B,
nat_plus: ℕ+,
assert: ↑b,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
nat_plus: ℕ+,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
sq_type: SQType(T),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True
Lemmas referenced :
rationals_wf,
biject_wf,
nat_plus_wf,
assert_wf,
is-qrep_wf,
qrep_wf,
assert-is-qrep,
istype-int,
product_subtype_base,
int_subtype_base,
set_subtype_base,
less_than_wf,
equals-qrep,
subtype_base_sq,
sq_stable__assert,
assert_elim,
bool_wf,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_pairFormation_alt,
lambdaEquality_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
productEquality,
intEquality,
dependent_functionElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
because_Cache,
productElimination,
independent_functionElimination,
equalityIsType4,
productIsType,
applyEquality,
sqequalRule,
independent_isectElimination,
lambdaFormation_alt,
natural_numberEquality,
inhabitedIsType,
independent_pairFormation,
setIsType,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mBbbQ{} \msim{} \{p:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| \muparrow{}is-qrep(p)\}
Date html generated:
2019_10_16-AM-11_47_49
Last ObjectModification:
2018_10_10-PM-01_24_37
Theory : rationals
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