Nuprl Lemma : qdiv_functionality_wrt_qle
∀[a,b,c,d:ℚ]. ((a/c) ≤ (b/d)) supposing ((c ≥ d) and (a ≤ b) and 0 < d and (0 ≤ a))
Proof
Definitions occuring in Statement :
qge: a ≥ b,
qle: r ≤ s,
qless: r < s,
qdiv: (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,
not: ¬A,
implies: P ⇒ Q,
qge: a ≥ b,
guard: {T},
subtype_rel: A ⊆r B,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qle_weakening_lt_qorder,
qmul_functionality_wrt_qle,
qmul_ac_1_qrng,
qmul_comm_qrng,
iff_weakening_equal,
qmul_wf,
qmul-qdiv-cancel,
true_wf,
squash_wf,
qmul_preserves_qle,
qless_wf,
qle_wf,
qge_wf,
rationals_wf,
equal_wf,
qless_irreflexivity,
qle_weakening_eq_qorder,
int-subtype-rationals,
qless_transitivity_2_qorder,
qdiv_wf,
qle_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
lambdaFormation,
hypothesis,
natural_numberEquality,
applyEquality,
sqequalRule,
hypothesisEquality,
voidElimination,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a,b,c,d:\mBbbQ{}]. ((a/c) \mleq{} (b/d)) supposing ((c \mgeq{} d) and (a \mleq{} b) and 0 < d and (0 \mleq{} a))
Date html generated:
2016_05_15-PM-11_04_29
Last ObjectModification:
2016_01_16-PM-09_28_33
Theory : rationals
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