Nuprl Lemma : assert-qeq
∀[r,s:ℚ]. uiff(↑qeq(r;s);r = s ∈ ℚ)
Proof
Definitions occuring in Statement :
rationals: ℚ,
qeq: qeq(r;s),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
rationals: ℚ,
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert_wf,
qeq_wf2,
assert_witness,
rationals_wf,
eqtt_to_assert,
quotient-member-eq,
equal-wf-T-base,
qeq-equiv,
equal_wf,
squash_wf,
true_wf,
istype-universe,
bool_wf,
subtype_rel_self,
iff_weakening_equal,
b-union_wf,
int_nzero_wf,
qeq_wf,
equal-wf-base,
qeq_refl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
equalityIsType1,
inhabitedIsType,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
productIsType,
equalityIsType4,
intEquality,
productEquality,
equalityIsType3,
applyLambdaEquality,
hyp_replacement,
lambdaEquality
Latex:
\mforall{}[r,s:\mBbbQ{}]. uiff(\muparrow{}qeq(r;s);r = s)
Date html generated:
2019_10_16-AM-11_47_40
Last ObjectModification:
2018_10_10-PM-01_24_41
Theory : rationals
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