Nuprl Lemma : rdiv_functionality_wrt_rleq2
∀[x,y,z,w:ℝ]. ((x/w) ≤ (z/y)) supposing ((x ≤ z) and (y ≤ w) and ((r0 < y) ∧ ((r0 ≤ x) ∨ (r0 ≤ z))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
rge: x ≥ y,
cand: A c∧ B
Lemmas referenced :
rmul_preserves_rleq,
rdiv_wf,
rless_transitivity1,
less_than'_wf,
rsub_wf,
rless_wf,
int-to-real_wf,
nat_plus_wf,
rleq_wf,
or_wf,
real_wf,
rmul_wf,
rinv_wf2,
uiff_transitivity,
rleq_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
rmul-rinv3,
rinv-mul-as-rdiv,
req_weakening,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmul_functionality_wrt_rleq2,
rleq_weakening_rless,
rleq_transitivity,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
hypothesis,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
hypothesisEquality,
independent_pairEquality,
applyEquality,
natural_numberEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
productEquality,
voidElimination,
computeAll,
int_eqEquality,
intEquality,
voidEquality,
unionElimination,
inlFormation,
independent_pairFormation
Latex:
\mforall{}[x,y,z,w:\mBbbR{}].
((x/w) \mleq{} (z/y)) supposing ((x \mleq{} z) and (y \mleq{} w) and ((r0 < y) \mwedge{} ((r0 \mleq{} x) \mvee{} (r0 \mleq{} z))))
Date html generated:
2017_10_03-AM-08_34_44
Last ObjectModification:
2017_04_07-AM-11_26_45
Theory : reals
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