Nuprl Lemma : qmul-qdiv
∀[a,b,c,d:ℚ]. (((a/c) * (b/d)) = (a * b/c * d) ∈ ℚ) supposing ((¬(d = 0 ∈ ℚ)) and (¬(c = 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,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
cand: A c∧ B,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
qmul-preserves-eq,
qmul_wf,
qdiv_wf,
not_wf,
equal-wf-T-base,
rationals_wf,
qmul-not-zero,
equal_wf,
squash_wf,
true_wf,
qmul_comm_qrng,
qmul-qdiv-cancel3,
iff_weakening_equal,
qmul_ac_1_qrng,
qmul-qdiv-cancel,
qmul-qdiv-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
productElimination,
baseClosed,
sqequalRule,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
independent_functionElimination
Latex:
\mforall{}[a,b,c,d:\mBbbQ{}]. (((a/c) * (b/d)) = (a * b/c * d)) supposing ((\mneg{}(d = 0)) and (\mneg{}(c = 0)))
Date html generated:
2018_05_21-PM-11_56_41
Last ObjectModification:
2017_07_26-PM-06_47_14
Theory : rationals
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