Nuprl Lemma : qdiv-qdiv
∀[a,b,c:ℚ]. ((a/(b/c)) = (a * c/b) ∈ ℚ) supposing ((¬(c = 0 ∈ ℚ)) and (¬(b = 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,
prop: ℙ,
qdiv: (r/s),
not: ¬A,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
false: False,
true: True,
squash: ↓T,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
equal-wf-T-base,
rationals_wf,
qinv-zero,
qmul-zero,
qinv_wf,
assert-qeq,
assert_wf,
qeq_wf2,
int-subtype-rationals,
qmul_wf,
or_wf,
equal_wf,
qdiv_wf,
squash_wf,
true_wf,
qmul_one_qrng,
qmul_comm_qrng,
iff_weakening_equal,
qmul-qdiv-cancel4,
qmul_assoc_qrng,
qmul-qdiv-cancel,
qmul_com,
qmul_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
baseClosed,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
addLevel,
impliesFunctionality,
dependent_functionElimination,
productElimination,
natural_numberEquality,
applyEquality,
independent_functionElimination,
lambdaFormation,
unionElimination,
voidElimination,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. ((a/(b/c)) = (a * c/b)) supposing ((\mneg{}(c = 0)) and (\mneg{}(b = 0)))
Date html generated:
2018_05_21-PM-11_58_41
Last ObjectModification:
2017_07_26-PM-06_48_17
Theory : rationals
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