Nuprl Lemma : qmul-qdiv-cancel6
∀[a,b,c:ℚ]. (((b/c * a) * a) = (b/c) ∈ ℚ) supposing ((¬(c = 0 ∈ ℚ)) and (¬(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,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
not: ¬A,
false: False
Lemmas referenced :
not_wf,
equal-wf-T-base,
rationals_wf,
qmul-preserves-eq,
qmul_wf,
qdiv_wf,
equal_wf,
squash_wf,
true_wf,
qmul_ac_1_qrng,
qmul_comm_qrng,
qmul-qdiv-cancel2,
qmul-qdiv-cancel,
iff_weakening_equal,
int-subtype-rationals,
qmul_zero_qrng,
qmul_one_qrng
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,
productElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
independent_functionElimination,
lambdaFormation,
applyLambdaEquality,
voidElimination
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. (((b/c * a) * a) = (b/c)) supposing ((\mneg{}(c = 0)) and (\mneg{}(a = 0)))
Date html generated:
2018_05_21-PM-11_51_13
Last ObjectModification:
2017_07_26-PM-06_44_23
Theory : rationals
Home
Index