Nuprl Lemma : qinv-positive
∀[v:ℚ]. 0 < (1/v) supposing 0 < v
Proof
Definitions occuring in Statement :
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,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
guard: {T},
false: False,
qdiv: (r/s),
true: True,
squash: ↓T,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(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,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
or: P ∨ Q
Lemmas referenced :
qless_transitivity,
qless-minus,
qmul_inv,
qmul_com,
iff_weakening_equal,
qmul_comm_qrng,
qmul_one_qrng,
qmul_wf,
true_wf,
squash_wf,
qinv_wf,
qless_wf,
qless_irreflexivity,
qle_weakening_eq_qorder,
qless_transitivity_2_qorder,
qless_witness,
qdiv_wf,
qmul-positive,
rationals_wf,
equal_wf,
not_wf,
int-subtype-rationals,
qeq_wf2,
assert_wf,
assert-qeq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
impliesFunctionality,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation,
voidElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
unionElimination,
minusEquality
Latex:
\mforall{}[v:\mBbbQ{}]. 0 < (1/v) supposing 0 < v
Date html generated:
2016_05_15-PM-11_03_53
Last ObjectModification:
2016_01_16-PM-09_28_59
Theory : rationals
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