Nuprl Lemma : proj-eq-iff
∀n:ℕ. ∀a,b:ℙ^n. (a = b ⇐⇒ ↓∃m:{m:ℝ| m ≠ r0} . req-vec(n + 1;a;m*b))
Proof
Definitions occuring in Statement :
proj-eq: a = b,
real-proj: ℙ^n,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
squash: ↓T,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
real-proj: ℙ^n,
proj-eq: a = b,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
le: A ≤ B,
real-vec: ℝ^n,
cand: A c∧ B,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P)
Lemmas referenced :
proj-eq_wf,
squash_wf,
real_wf,
rneq_wf,
int-to-real_wf,
req-vec_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
real-vec-mul_wf,
real-proj_wf,
istype-nat,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
rdiv_wf,
rmul-nonzero,
rmul_wf,
rneq_functionality,
req_weakening,
rdiv-nonzero,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
req_functionality,
rmul_comm,
req-same,
req_transitivity,
rmul_functionality,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
sq_stable__proj-eq,
int_seg_wf,
rmul_assoc,
rmul-ac
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
introduction,
cut,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
universeIsType,
extract_by_obid,
isectElimination,
productEquality,
setEquality,
natural_numberEquality,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
voidElimination,
because_Cache,
inhabitedIsType,
productElimination,
applyEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbP{}\^{}n. (a = b \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}m:\{m:\mBbbR{}| m \mneq{} r0\} . req-vec(n + 1;a;m*b))
Date html generated:
2020_05_20-PM-01_16_43
Last ObjectModification:
2020_01_06-PM-00_10_31
Theory : inner!product!spaces
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