Nuprl Lemma : qmul-qdiv-cancel
∀[a,b:ℚ]. (a * (b/a)) = b ∈ ℚ supposing ¬(a = 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,
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,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
qdiv: (r/s),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-T-base,
rationals_wf,
qinv_wf,
equal_wf,
qmul_assoc,
iff_weakening_equal,
qmul_inv_l,
qmul_wf,
squash_wf,
true_wf,
qmul_comm_qrng,
qmul_ac_1_qrng,
qmul_one_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
impliesFunctionality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
sqequalRule,
productElimination,
independent_isectElimination,
because_Cache,
baseClosed,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
imageMemberEquality,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
universeEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. (a * (b/a)) = b supposing \mneg{}(a = 0)
Date html generated:
2018_05_21-PM-11_50_48
Last ObjectModification:
2017_07_26-PM-06_44_09
Theory : rationals
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