Nuprl Lemma : rmul_preserves_req
∀[x,y,z:ℝ]. uiff(x = z;(x * y) = (z * y)) supposing y ≠ r0
Proof
Definitions occuring in Statement :
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,
rev_uimplies: rev_uimplies(P;Q),
implies: P ⇒ Q,
prop: ℙ,
rdiv: (x/y),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
req_functionality,
rmul_wf,
rmul_functionality,
req_weakening,
req_witness,
req_wf,
rneq_wf,
int-to-real_wf,
real_wf,
rdiv_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req_transitivity,
rmul-rinv,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
universeIsType,
sqequalRule,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
natural_numberEquality,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination
Latex:
\mforall{}[x,y,z:\mBbbR{}]. uiff(x = z;(x * y) = (z * y)) supposing y \mneq{} r0
Date html generated:
2019_10_29-AM-09_40_11
Last ObjectModification:
2019_04_01-PM-07_01_17
Theory : reals
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