Nuprl Lemma : non-neg-qmul
∀[a,b:ℚ]. (0 ≤ (a * b)) supposing ((0 ≤ b) and (0 ≤ a))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
guard: {T},
iff: P ⇐⇒ Q,
cand: A c∧ B,
true: True,
squash: ↓T,
rev_implies: P ⇐ Q
Lemmas referenced :
qle_witness,
int-subtype-rationals,
qmul_wf,
qle_wf,
rationals_wf,
or_wf,
qless_wf,
equal-wf-base-T,
qle-iff,
qmul-positive,
equal_wf,
squash_wf,
true_wf,
qmul_zero_qrng,
iff_weakening_equal,
member_wf,
qmul_comm_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
independent_functionElimination,
hypothesis,
extract_by_obid,
isectElimination,
natural_numberEquality,
applyEquality,
sqequalRule,
hypothesisEquality,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
unionElimination,
baseClosed,
addLevel,
impliesFunctionality,
dependent_functionElimination,
productElimination,
independent_isectElimination,
functionEquality,
inlFormation,
inrFormation,
independent_pairFormation,
productEquality,
minusEquality,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
instantiate
Latex:
\mforall{}[a,b:\mBbbQ{}]. (0 \mleq{} (a * b)) supposing ((0 \mleq{} b) and (0 \mleq{} a))
Date html generated:
2018_05_21-PM-11_56_08
Last ObjectModification:
2017_07_26-PM-06_46_43
Theory : rationals
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