Nuprl Lemma : rv-five-segment
∀[n:ℕ]. ∀[a,b,c,d,A,B,C,D:ℝ^n]. (cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and A-B-C and a-b-c)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
rv-congruent: ab=cd,
real-vec: ℝ^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rv-congruent: ab=cd,
rv-between: a-b-c,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
let: let,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
cand: A c∧ B,
real-vec-sep: a ≠ b,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
req_witness,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
real-vec-between_wf,
real-vec-sep_wf,
real-vec_wf,
nat_wf,
rsqrt_wf,
rnexp2-nonneg,
rnexp_wf,
false_wf,
le_wf,
square-nonneg,
rmul_wf,
req_weakening,
rsqrt_functionality,
uiff_transitivity,
req_functionality,
rnexp2,
rsqrt-of-square,
rnexp_functionality,
real-vec-dist-symmetry,
rv-five-segment-lemma,
member_rooint_lemma,
radd-preserves-rless,
rsub_wf,
rless_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_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,
rdiv_wf,
rless_wf,
req_transitivity,
radd_functionality,
rmul_functionality,
rsub_functionality,
real-vec-dist-between-2,
real-vec-add_wf,
real-vec-mul_wf,
rabs_wf,
real-vec-dist_functionality,
req-vec_weakening,
req-vec_inversion,
req_inversion,
req-vec_wf,
rooint_wf,
i-member_wf,
real-vec-dist-between,
rleq_weakening_rless,
rabs-of-nonneg,
set_wf,
equal_wf,
rmul_preserves_req,
rdiv_functionality
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
dependent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
independent_isectElimination,
voidElimination,
voidEquality,
computeAll,
int_eqEquality,
intEquality,
inlFormation,
dependent_pairFormation,
inrFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b,c,d,A,B,C,D:\mBbbR{}\^{}n].
(cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and A-B-C and a-b-c)
Date html generated:
2017_10_03-AM-11_19_51
Last ObjectModification:
2017_07_28-AM-08_26_15
Theory : reals
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