Nuprl Lemma : rv-inner-Pasch
∀n:ℕ. ∀a,b,c,p,q:ℝ^n. (a-p-c ⇒ b-q-c ⇒ (∃x:ℝ^n. ((a ≠ q ⇒ a-x-q) ∧ (b ≠ p ⇒ b-x-p))))
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rneq: x ≠ y,
or: P ∨ Q,
rdiv: (x/y),
true: True,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
subtype_rel: A ⊆r B,
cand: A c∧ B
Lemmas referenced :
rv-between_wf,
real-vec_wf,
istype-nat,
member_rooint_lemma,
istype-void,
rmul_preserves_rless,
int-to-real_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
radd-preserves-rless,
rsub_wf,
radd_wf,
itermAdd_wf,
rless_transitivity2,
rleq_weakening_rless,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
req_weakening,
rdiv_wf,
rless_wf,
rminus_wf,
rinv_wf2,
itermMinus_wf,
rless-implies-rless,
req_transitivity,
radd_functionality,
rminus_functionality,
rmul-rinv3,
real_term_value_minus_lemma,
rmul_preserves_req,
iff_weakening_equal,
subtype_rel_self,
req_wf,
squash_wf,
true_wf,
real_wf,
req_functionality,
rmul_functionality,
rmul-rinv,
radd-preserves-req,
req_inversion,
real-vec-add_wf,
real-vec-mul_wf,
real-vec-sep_wf,
i-member_wf,
rooint_wf,
real-vec-between_functionality,
req-vec_weakening,
req-vec_wf,
req-vec_functionality,
real-vec-add_functionality,
req-vec_transitivity,
real-vec-mul-linear,
real-vec-mul-mul,
real-vec-mul_functionality,
req-vec_inversion,
real-vec-add-assoc,
real-vec-add-com
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
rename,
natural_numberEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
inrFormation_alt,
closedConclusion,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
dependent_pairFormation_alt,
productIsType,
functionIsType
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c,p,q:\mBbbR{}\^{}n. (a-p-c {}\mRightarrow{} b-q-c {}\mRightarrow{} (\mexists{}x:\mBbbR{}\^{}n. ((a \mneq{} q {}\mRightarrow{} a-x-q) \mwedge{} (b \mneq{} p {}\mRightarrow{} b-x-p))))
Date html generated:
2019_10_30-AM-08_51_25
Last ObjectModification:
2018_11_21-AM-09_59_45
Theory : reals
Home
Index