Nuprl Lemma : rv-congruent-preserves-sep
∀n:ℕ. ∀a,b,c,d:ℝ^n. (ab=cd ⇒ a ≠ b ⇒ c ≠ d)
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-congruent: ab=cd,
real-vec-sep: a ≠ b,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
real-vec-sep_wf,
rv-congruent_wf,
real-vec_wf,
nat_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
rless_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. (ab=cd {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} c \mneq{} d)
Date html generated:
2016_10_26-AM-10_31_11
Last ObjectModification:
2016_09_25-PM-01_13_28
Theory : reals
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