Nuprl Lemma : qinv-zero
∀[c:ℚ]. ¬(1/c = 0 ∈ ℚ) supposing ¬(c = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qinv: 1/r,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
qdiv: (r/s),
prop: ℙ,
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),
qmul: r * s,
btrue: tt,
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
lt_int: i <z j,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
bnot: ¬bb,
uiff: uiff(P;Q),
and: P ∧ Q
Lemmas referenced :
qless_trichot_qorder,
qinv-negative,
qless_wf,
qmul_wf,
qinv-positive,
equal-wf-T-base,
rationals_wf,
qinv_wf,
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
unionElimination,
isectElimination,
independent_isectElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
voidElimination,
independent_functionElimination,
addLevel,
impliesFunctionality,
productElimination,
baseClosed,
lambdaEquality,
isect_memberEquality,
equalityTransitivity
Latex:
\mforall{}[c:\mBbbQ{}]. \mneg{}(1/c = 0) supposing \mneg{}(c = 0)
Date html generated:
2016_10_26-AM-06_32_48
Last ObjectModification:
2016_07_12-AM-07_56_12
Theory : rationals
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