Nuprl Lemma : rv-congruent_functionality
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2,d1,d2:ℝ^n.
(req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ req-vec(n;d1;d2) ⇒ (a1b1=c1d1 ⇐⇒ a2b2=c2d2))
Proof
Definitions occuring in Statement :
rv-congruent: ab=cd,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rv-congruent: ab=cd,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q)
Lemmas referenced :
rv-congruent_wf,
req-vec_wf,
real-vec_wf,
nat_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_functionality,
real-vec-dist_functionality,
req-vec_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
productElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a1,a2,b1,b2,c1,c2,d1,d2:\mBbbR{}\^{}n.
(req-vec(n;a1;a2)
{}\mRightarrow{} req-vec(n;b1;b2)
{}\mRightarrow{} req-vec(n;c1;c2)
{}\mRightarrow{} req-vec(n;d1;d2)
{}\mRightarrow{} (a1b1=c1d1 \mLeftarrow{}{}\mRightarrow{} a2b2=c2d2))
Date html generated:
2016_10_26-AM-10_28_49
Last ObjectModification:
2016_09_25-PM-01_06_53
Theory : reals
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