Nuprl Lemma : qmul-preserves-eq
∀[a,b,c:ℚ]. uiff(a = b ∈ ℚ;(c * a) = (c * b) ∈ ℚ) supposing ¬(c = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qmul: r * s,
rationals: ℚ,
uiff: uiff(P;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,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Lemmas referenced :
and_wf,
equal_wf,
rationals_wf,
qmul_wf,
not_wf,
equal-wf-T-base,
qdiv_wf,
int-subtype-rationals,
squash_wf,
true_wf,
qmul_comm_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel2,
qmul_one_qrng,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyLambdaEquality,
setElimination,
rename,
productElimination,
hyp_replacement,
because_Cache,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
baseClosed,
natural_numberEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
independent_functionElimination
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. uiff(a = b;(c * a) = (c * b)) supposing \mneg{}(c = 0)
Date html generated:
2018_05_21-PM-11_51_03
Last ObjectModification:
2017_07_26-PM-06_44_18
Theory : rationals
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