Nuprl Lemma : rv-between-translation
∀n:ℕ. ∀a,b,c,z:ℝ^n. (z + a-z + b-z + c ⇐⇒ a-b-c)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-add: X + Y,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rv-between: a-b-c,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
member: t ∈ T,
req-vec: req-vec(n;x;y),
real-vec-add: X + Y,
real-vec-mul: a*X,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
cand: A c∧ B,
real-vec-sep: a ≠ b,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
real-vec: ℝ^n,
uiff: uiff(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rv-between_wf,
real-vec_wf,
nat_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
rless_functionality,
req_weakening,
real-vec-dist-translation2,
radd-preserves-req,
radd_wf,
rmul_wf,
rminus_wf,
req-implies-req,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermMinus_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
radd_functionality,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
promote_hyp,
dependent_pairFormation,
hypothesisEquality,
hypothesis,
cut,
dependent_functionElimination,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
productEquality,
applyEquality,
lambdaEquality,
setEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,z:\mBbbR{}\^{}n. (z + a-z + b-z + c \mLeftarrow{}{}\mRightarrow{} a-b-c)
Date html generated:
2017_10_03-AM-11_10_40
Last ObjectModification:
2017_07_28-AM-08_23_20
Theory : reals
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