Nuprl Lemma : equals-qrep
∀[r:ℚ]. (qrep(r) = r ∈ ℚ)
Proof
Definitions occuring in Statement :
qrep: qrep(r),
rationals: ℚ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
rationals: ℚ,
so_lambda: λ2x y.t[x; y],
guard: {T},
so_apply: x[s1;s2],
uimplies: b supposing a,
implies: P ⇒ Q,
pi2: snd(t),
so_apply: x[s],
so_lambda: λ2x.t[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
tunion: ⋃x:A.B[x],
b-union: A ⋃ B,
istype: istype(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
quotient: x,y:A//B[x; y]
Lemmas referenced :
qeq_wf,
qeq-qrep,
subtype_quotient,
equal-wf-T-base,
bool_wf,
qeq-equiv,
qrep_wf,
quotient-member-eq,
b-union_wf,
int_nzero_wf,
rationals_wf,
ifthenelse_wf,
nat_plus_inc_int_nzero,
istype-int,
nat_plus_wf,
subtype_rel_product,
bfalse_wf,
quotient_wf,
equal_functionality_wrt_subtype_rel2,
subtype_rel_self,
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
equal_wf,
qeq_refl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
lambdaFormation_alt,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
lambdaEquality_alt,
hypothesis,
baseClosed,
inhabitedIsType,
independent_isectElimination,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
intEquality,
productEquality,
universeEquality,
instantiate,
equalityTransitivity,
dependent_pairEquality_alt,
imageMemberEquality,
closedConclusion,
natural_numberEquality,
imageElimination,
productIsType,
productElimination,
pertypeElimination,
pointwiseFunctionality,
sqequalBase,
equalityIstype,
promote_hyp
Latex:
\mforall{}[r:\mBbbQ{}]. (qrep(r) = r)
Date html generated:
2019_10_16-AM-11_47_44
Last ObjectModification:
2019_06_25-PM-00_20_49
Theory : rationals
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