Nuprl Lemma : ip-congruent-same
∀[rv:InnerProductSpace]. ∀[a,b,c:Point]. (aa=bc ⇒ b ≡ c)
Proof
Definitions occuring in Statement :
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
ip-congruent: ab=cd,
prop: ℙ,
ss-eq: x ≡ y,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
rv-sub: x - y,
rv-minus: -x,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ip-congruent_wf,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-point_wf,
ss-eq_wf,
rv-sub_wf,
rv-0_wf,
rv-mul_wf,
int-to-real_wf,
radd_wf,
rv-add_wf,
req_wf,
rv-norm_wf,
real_wf,
rleq_wf,
rmul_wf,
rv-ip_wf,
uiff_transitivity,
ss-eq_functionality,
rv-mul-1-add,
ss-eq_weakening,
rv-mul_functionality,
radd-int,
rv-mul0,
req_functionality,
rv-norm_functionality,
req_weakening,
rv-norm0,
rv-norm-is-zero,
req_inversion,
rv-sub-is-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
isect_memberEquality,
voidElimination,
natural_numberEquality,
minusEquality,
setElimination,
rename,
setEquality,
productEquality,
independent_functionElimination,
productElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c:Point]. (aa=bc {}\mRightarrow{} b \mequiv{} c)
Date html generated:
2017_10_04-PM-11_56_43
Last ObjectModification:
2017_03_11-PM-02_49_10
Theory : inner!product!spaces
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