Nuprl Lemma : clear-denominator1
∀[a,b,c,d:ℝ]. uiff(((a/b) * c) = d;(c * a) = (d * b)) supposing b ≠ 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,
uiff: uiff(P;Q),
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,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
req_wf,
rdiv_wf,
rneq_wf,
int-to-real_wf,
real_wf,
rmul_preserves_req,
rinv_wf2,
req-implies-req,
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,
rsub_wf,
req_functionality,
req_transitivity,
rmul_functionality,
rmul-rinv3,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
dependent_functionElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[a,b,c,d:\mBbbR{}]. uiff(((a/b) * c) = d;(c * a) = (d * b)) supposing b \mneq{} r0
Date html generated:
2017_10_03-AM-08_38_24
Last ObjectModification:
2017_07_28-AM-07_30_36
Theory : reals
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