Nuprl Lemma : qmul_functionality_wrt_qless2
∀[a,b,c,d:ℚ]. (a * c < b * d) supposing ((c ≤ d) and a < b and (0 ≤ b) and 0 < c)
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: 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,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
squash: ↓T,
iff: P ⇐⇒ Q
Lemmas referenced :
iff_weakening_equal,
qmul_comm_qrng,
true_wf,
squash_wf,
qless_transitivity_2_qorder,
rationals_wf,
int-subtype-rationals,
qle_wf,
qless_wf,
qless_witness,
qmul_preserves_qless,
qmul_wf,
qle_witness,
qmul_preserves_qle2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
independent_functionElimination,
because_Cache,
productElimination,
independent_pairFormation,
sqequalRule,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a,b,c,d:\mBbbQ{}]. (a * c < b * d) supposing ((c \mleq{} d) and a < b and (0 \mleq{} b) and 0 < c)
Date html generated:
2016_05_15-PM-11_00_46
Last ObjectModification:
2016_01_16-PM-09_30_37
Theory : rationals
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