Nuprl Lemma : another-test-ring-req
∀a,b,c,d,e,x:ℝ. (b ≠ r0 ⇒ d ≠ r0 ⇒ x ≠ r0 ⇒ (((a/b) * (c/d) * (b * e/x)) = ((a * c/d) * e/x)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rneq_wf,
int-to-real_wf,
real_wf,
req_wf,
rmul_wf,
rinv_wf2,
rdiv_wf,
req_weakening,
uiff_transitivity,
req_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_functionality,
rmul-rinv3,
rinv-mul-as-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
independent_functionElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
dependent_functionElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination
Latex:
\mforall{}a,b,c,d,e,x:\mBbbR{}. (b \mneq{} r0 {}\mRightarrow{} d \mneq{} r0 {}\mRightarrow{} x \mneq{} r0 {}\mRightarrow{} (((a/b) * (c/d) * (b * e/x)) = ((a * c/d) * e/x)))
Date html generated:
2017_10_03-AM-08_34_52
Last ObjectModification:
2017_07_28-AM-07_28_39
Theory : reals
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