Nuprl Lemma : qdiv-non-neg1
∀[a,b:ℚ]. 0 ≤ (a/b) supposing 0 < b ∧ (0 ≤ a)
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qdiv: (r/s),
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
not: ¬A,
implies: P ⇒ Q,
guard: {T},
false: False,
prop: ℙ,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
iff_weakening_equal,
qmul-qdiv-cancel,
qmul_zero_qrng,
true_wf,
squash_wf,
qmul_wf,
qle_wf,
qless_wf,
and_wf,
rationals_wf,
equal_wf,
qless_irreflexivity,
qle_weakening_eq_qorder,
qless_transitivity_2_qorder,
int-subtype-rationals,
qle_witness,
qdiv_wf,
qmul_preserves_qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
natural_numberEquality,
applyEquality,
sqequalRule,
lambdaFormation,
hypothesisEquality,
voidElimination,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. 0 \mleq{} (a/b) supposing 0 < b \mwedge{} (0 \mleq{} a)
Date html generated:
2016_05_15-PM-11_05_04
Last ObjectModification:
2016_01_16-PM-09_28_00
Theory : rationals
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