Nuprl Lemma : qmul_preserves_qless
∀[a,b,c:ℚ]. uiff(a < b;c * a < c * b) supposing 0 < c
Proof
Definitions occuring in Statement :
qless: r < s,
qmul: r * s,
rationals: ℚ,
uiff: uiff(P;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,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
qsub: r - s,
all: ∀x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
not: ¬A,
false: False
Lemmas referenced :
qless_witness,
qmul_wf,
qless_wf,
int-subtype-rationals,
rationals_wf,
qadd_preserves_qless,
qadd_wf,
squash_wf,
true_wf,
qmul_comm_qrng,
qadd_comm_q,
qinverse_q,
iff_weakening_equal,
equal_wf,
qsub_wf,
qadd_com,
qmul_assoc_qrng,
q_distrib,
qmul-positive,
qadd_inv_assoc_q,
mon_ident_q,
uiff_transitivity2,
qminus_positive,
assert-qpositive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
minusEquality,
independent_isectElimination,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
hyp_replacement,
applyLambdaEquality,
dependent_functionElimination,
inlFormation,
productEquality,
unionElimination,
lambdaFormation,
promote_hyp,
voidElimination
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. uiff(a < b;c * a < c * b) supposing 0 < c
Date html generated:
2018_05_21-PM-11_56_20
Last ObjectModification:
2017_07_26-PM-06_46_51
Theory : rationals
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