Nuprl Lemma : qmul_reverses_qle
∀[a,b,c:ℚ]. uiff(a ≤ b;(c * b) ≤ (c * a)) supposing c < 0
Proof
Definitions occuring in Statement :
qle: r ≤ s,
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,
subtype_rel: A ⊆r B,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
true: True,
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qmul_preserves_qle,
qmul_com,
qadd_ac_1_q,
qmul_comm_qrng,
qmul_over_minus_qrng,
qadd_preserves_qle,
qle_weakening_lt_qorder,
qmul_preserves_qle2,
iff_weakening_equal,
mon_ident_q,
qinverse_q,
qadd_comm_q,
true_wf,
squash_wf,
qle_witness,
rationals_wf,
int-subtype-rationals,
qless_wf,
qle_wf,
qadd_wf,
qmul_wf,
qadd_preserves_qless
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
because_Cache,
minusEquality,
natural_numberEquality,
hypothesis,
applyEquality,
sqequalRule,
hypothesisEquality,
productElimination,
independent_isectElimination,
isect_memberFormation,
introduction,
independent_pairEquality,
isect_memberEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_pairFormation
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. uiff(a \mleq{} b;(c * b) \mleq{} (c * a)) supposing c < 0
Date html generated:
2016_05_15-PM-10_59_36
Last ObjectModification:
2016_01_16-PM-09_31_58
Theory : rationals
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