Nuprl Lemma : ip-congruent_functionality
∀[rv:InnerProductSpace]. ∀[a,b,c,d,a2,b2,c2,d2:Point].
({ab=cd ⇐⇒ a2b2=c2d2}) supposing (d ≡ d2 and c ≡ c2 and b ≡ b2 and a ≡ a2)
Proof
Definitions occuring in Statement :
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
iff: P ⇐⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
ip-congruent: ab=cd,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ip-congruent_wf,
req_witness,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
ss-eq_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-point_wf,
req_functionality,
rv-norm_functionality,
rv-sub_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
because_Cache,
independent_functionElimination,
instantiate,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c,d,a2,b2,c2,d2:Point].
(\{ab=cd \mLeftarrow{}{}\mRightarrow{} a2b2=c2d2\}) supposing (d \mequiv{} d2 and c \mequiv{} c2 and b \mequiv{} b2 and a \mequiv{} a2)
Date html generated:
2017_10_04-PM-11_56_39
Last ObjectModification:
2017_03_09-PM-05_36_19
Theory : inner!product!spaces
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