Nuprl Lemma : qdiv-qminus
∀[x,y:ℚ]. (x/-(y)) = (-(x)/y) ∈ ℚ supposing ¬(y = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
minus: -n,
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,
subtype_rel: A ⊆r B,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
qdiv: (r/s),
qmul: r * s,
qinv: 1/r,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qmul_wf,
equal-wf-T-base,
int-subtype-rationals,
qmul-preserves-eq,
qdiv_wf,
assert-qeq,
equal-wf-base,
equal_wf,
qmul-qdiv,
iff_weakening_equal,
rationals_wf,
not_wf,
squash_wf,
true_wf,
qmul_zero_qrng,
qinv_inv_q,
qmul_over_minus_qrng,
qmul_one_qrng,
qmul_comm_qrng,
qmul-qdiv-cancel2,
qmul-qdiv-cancel,
qmul_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
independent_functionElimination,
thin,
applyLambdaEquality,
extract_by_obid,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
hypothesisEquality,
voidElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairFormation,
baseClosed,
lambdaEquality,
imageElimination,
imageMemberEquality,
hyp_replacement,
isect_memberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[x,y:\mBbbQ{}]. (x/-(y)) = (-(x)/y) supposing \mneg{}(y = 0)
Date html generated:
2018_05_21-PM-11_56_46
Last ObjectModification:
2017_07_26-PM-06_47_18
Theory : rationals
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