Nuprl Lemma : qmul_functionality_wrt_qle
∀[a,b,c,d:ℚ]. ((a * c) ≤ (b * d)) supposing ((c ≤ d) and (a ≤ b) and (0 ≤ c) and (0 ≤ a))
Proof
Definitions occuring in Statement :
qle: 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,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
not: ¬A,
false: False,
qle: r ≤ s,
grp_leq: a ≤ b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j)
Lemmas referenced :
decidable__equal_rationals,
qle_witness,
qmul_wf,
qle_wf,
int-subtype-rationals,
rationals_wf,
squash_wf,
true_wf,
qmul_zero_qrng,
iff_weakening_equal,
qmul-non-neg,
or_wf,
equal-wf-T-base,
qless_wf,
qle_transitivity_qorder,
qle-iff,
qle_antisymmetry,
qle_weakening_eq_qorder,
qmul_preserves_qle2,
qmul_comm_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
unionElimination,
isectElimination,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
hyp_replacement,
Error :applyLambdaEquality,
inlFormation,
productEquality,
minusEquality,
inrFormation,
independent_pairFormation,
promote_hyp,
equalityElimination,
voidElimination
Latex:
\mforall{}[a,b,c,d:\mBbbQ{}]. ((a * c) \mleq{} (b * d)) supposing ((c \mleq{} d) and (a \mleq{} b) and (0 \mleq{} c) and (0 \mleq{} a))
Date html generated:
2016_10_25-PM-00_07_50
Last ObjectModification:
2016_07_12-AM-07_50_49
Theory : rationals
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