Nuprl Lemma : rv-T-partially-implies-rv-T'
∀n:ℕ+. ∀a,b,c:ℝ^n. ((a ≠ c ∨ (¬a ≠ c)) ⇒ rv-T(n;a;b;c) ⇒ rv-T'(n;a;b;c))
Proof
Definitions occuring in Statement :
rv-T': rv-T'(n;a;b;c),
rv-T: rv-T(n;a;b;c),
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-T': rv-T'(n;a;b;c),
or: P ∨ Q,
rv-T: rv-T(n;a;b;c),
and: P ∧ Q,
real-vec-be: real-vec-be(n;a;b;c),
exists: ∃x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
rv-between: a-b-c,
real-vec-between: a-b-c,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
real: ℝ,
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
nat_plus: ℕ+
Lemmas referenced :
member_rccint_lemma,
not-real-vec-sep-iff-eq,
nat_plus_subtype_nat,
rv-between_wf,
rv-T_wf,
or_wf,
real-vec-sep_wf,
not_wf,
real-vec_wf,
nat_plus_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
int-to-real_wf,
rv-between_functionality,
req-vec_weakening,
real-vec-add_functionality,
real-vec-mul_functionality,
req_weakening,
radd_wf,
rmul_wf,
i-member_wf,
rooint_wf,
req-vec_wf,
member_rooint_lemma,
rless-cases,
radd-zero-both,
radd-rminus-both,
radd_functionality,
radd-ac,
radd_comm,
rleq_functionality,
uiff_transitivity,
rmul-zero-both,
rless_functionality,
rminus_wf,
rleq_wf,
radd-preserves-rleq,
rmul_preserves_rless,
rmul_functionality_wrt_rleq2,
radd_functionality_wrt_rleq,
rless_functionality_wrt_implies,
rless_wf,
rleq_weakening_rless,
rleq_weakening_equal,
rless_transitivity2,
radd-preserves-rless,
rmul-nonneg-case1,
equal_wf,
real_wf,
rmul_comm,
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,
rleq-rmax,
rmul_preserves_rleq2,
rmax_wf,
less_than'_wf,
itermMultiply_wf,
real_term_value_mul_lemma,
rleq_functionality_wrt_implies,
req_inversion,
rmul-distrib1,
req_wf,
rleq_weakening,
rmul_functionality,
rmax_strict_lb,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
independent_functionElimination,
hypothesis,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
independent_isectElimination,
because_Cache,
natural_numberEquality,
independent_pairFormation,
dependent_pairFormation,
productEquality,
levelHypothesis,
addLevel,
equalitySymmetry,
equalityTransitivity,
inlFormation,
promote_hyp,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberFormation,
independent_pairEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}a,b,c:\mBbbR{}\^{}n. ((a \mneq{} c \mvee{} (\mneg{}a \mneq{} c)) {}\mRightarrow{} rv-T(n;a;b;c) {}\mRightarrow{} rv-T'(n;a;b;c))
Date html generated:
2017_10_03-AM-11_25_42
Last ObjectModification:
2017_07_28-AM-08_27_07
Theory : reals
Home
Index