Nuprl Lemma : rmul-rdiv2
∀[x,a,b:ℝ]. ((x/a * b) = ((x/a) * (r1/b))) supposing (b ≠ r0 and a ≠ r0)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
false: False,
implies: P ⇒ Q,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
and: P ∧ Q,
true: True,
all: ∀x:A. B[x],
pi2: snd(t),
prop: ℙ
Lemmas referenced :
assert-rat-term-eq2,
rtermDivide_wf,
rtermVar_wf,
rtermMultiply_wf,
rtermConstant_wf,
int-to-real_wf,
istype-int,
rmul-neq-zero,
req_witness,
rdiv_wf,
rmul_wf,
rneq_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
lambdaEquality_alt,
int_eqEquality,
hypothesisEquality,
independent_isectElimination,
approximateComputation,
sqequalRule,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[x,a,b:\mBbbR{}]. ((x/a * b) = ((x/a) * (r1/b))) supposing (b \mneq{} r0 and a \mneq{} r0)
Date html generated:
2019_10_29-AM-09_57_10
Last ObjectModification:
2019_04_01-PM-07_10_28
Theory : reals
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