Nuprl Lemma : rv-extend
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a ≠ b ⇒ (∃x:{x:ℝ^n| bx=cd} . (c ≠ d ⇒ a-b-x)))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec-between: a-b-c,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
rv-congruent: ab=cd,
req: x = y,
prop: ℙ,
real-vec-sep: a ≠ b
Lemmas referenced :
rv-extend-1,
real-vec-dist_wf,
rv-congruent_wf,
real-vec-sep_wf,
real-vec-between_wf,
real-vec_wf,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
functionEquality,
setElimination,
rename,
because_Cache
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. (a \mneq{} b {}\mRightarrow{} (\mexists{}x:\{x:\mBbbR{}\^{}n| bx=cd\} . (c \mneq{} d {}\mRightarrow{} a-b-x)))
Date html generated:
2016_10_26-AM-10_39_48
Last ObjectModification:
2016_09_25-PM-11_50_49
Theory : reals
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