Nuprl Lemma : rv-tarski-parallel
∀n:ℕ. ∀a,b,c:ℝ^n. (a ≠ b ⇒ a ≠ c ⇒ (∀d,p:ℝ^n. (b-d-c ⇒ a-d-p ⇒ (∃x,y:ℝ^n. (a-b-x ∧ x-p-y ∧ a-c-y)))))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-between: a-b-c,
and: P ∧ Q,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
i-member: r ∈ I,
rooint: (l, u),
real-vec-mul: a*X,
real-vec-sub: X - Y,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
subtype_rel: A ⊆r B,
real-vec-sep: a ≠ b,
rge: x ≥ y,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
rsub: x - y
Lemmas referenced :
member_rooint_lemma,
radd-preserves-rless,
int-to-real_wf,
rsub_wf,
rless_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
real-vec-mul_wf,
real-vec-sub_wf,
rdiv_wf,
rless_wf,
rv-between_wf,
exists_wf,
real-vec_wf,
real-vec-sep_wf,
nat_wf,
real_wf,
equal_wf,
i-member_wf,
rooint_wf,
req-vec_wf,
real-vec-add_wf,
int_seg_wf,
rmul_wf,
rminus_wf,
rinv_wf2,
itermMultiply_wf,
itermMinus_wf,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
req_functionality,
req_weakening,
radd_functionality,
rmul-assoc,
req_transitivity,
rminus_functionality,
rmul_functionality,
rmul-rinv,
real-vec-dist_wf,
real-vec-dist-between,
real-vec-dist-nonneg,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rless_functionality_wrt_implies,
trivial-rless-radd,
rleq_wf,
rminus-rminus,
rmul-ac,
real-vec-sub_functionality,
req-vec_weakening,
req-vec_functionality,
radd-rminus-assoc,
radd_comm,
radd-ac,
radd-assoc,
req_inversion,
rmul_comm,
rmul-one-both,
rmul_over_rminus,
rmul-distrib,
uiff_transitivity,
rsub_functionality,
req_wf,
rmul_preserves_req,
req-implies-req,
rmul-rinv3,
rabs_wf,
real-vec-dist-dilation,
real-vec-dist-translation,
rmul_preserves_rless,
rabs_functionality,
rmul-identity1,
rinv-as-rdiv,
rdiv_functionality,
radd-zero-both,
rminus-zero,
rmul-zero-both,
rmul-rdiv-cancel2,
rleq_functionality,
rabs-of-nonneg,
rleq_weakening_rless,
rless_transitivity2,
rmul_preserves_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
rename,
isectElimination,
natural_numberEquality,
hypothesisEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_pairFormation,
inrFormation,
independent_pairFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
applyEquality,
setEquality,
levelHypothesis,
addLevel
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n.
(a \mneq{} b {}\mRightarrow{} a \mneq{} c {}\mRightarrow{} (\mforall{}d,p:\mBbbR{}\^{}n. (b-d-c {}\mRightarrow{} a-d-p {}\mRightarrow{} (\mexists{}x,y:\mBbbR{}\^{}n. (a-b-x \mwedge{} x-p-y \mwedge{} a-c-y)))))
Date html generated:
2017_10_03-AM-11_18_30
Last ObjectModification:
2017_07_28-AM-08_25_40
Theory : reals
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