Nuprl Lemma : rv-ip-rneq
∀rv:InnerProductSpace. ∀a1,b1,a2,b2:Point(rv). (a1 ⋅ b1 ≠ a2 ⋅ b2 ⇒ (a1 # a2 ∨ b1 # b2))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rneq: x ≠ y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rneq-cases,
rv-ip_wf,
Error :ss-sep_wf,
rneq_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rv-ip-rneq-0,
rv-sub_wf,
rv-sep-iff,
rsub_wf,
int-to-real_wf,
rneq-by-function,
rminus_wf,
real_wf,
req-implies-req,
req_wf,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
itermConstant_wf,
rneq_functionality,
rv-ip-sub,
req_weakening,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
Error :ss-sep-symmetry,
rv-ip-sub2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
unionElimination,
inlFormation_alt,
universeIsType,
applyEquality,
because_Cache,
sqequalRule,
inrFormation_alt,
inhabitedIsType,
instantiate,
independent_isectElimination,
productElimination,
natural_numberEquality,
lambdaEquality_alt,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a1,b1,a2,b2:Point(rv). (a1 \mcdot{} b1 \mneq{} a2 \mcdot{} b2 {}\mRightarrow{} (a1 \# a2 \mvee{} b1 \# b2))
Date html generated:
2020_05_20-PM-01_11_44
Last ObjectModification:
2019_12_08-PM-07_01_58
Theory : inner!product!spaces
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