Nuprl Lemma : not-proj-sep-iff-proj-eq
∀n:ℕ. ∀a,b:ℙ^n. (¬a ≠ b ⇐⇒ a = b)
Proof
Definitions occuring in Statement :
proj-eq: a = b,
proj-sep: a ≠ b,
real-proj: ℙ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
false: False,
rev_implies: P ⇐ Q,
or: P ∨ Q,
squash: ↓T,
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
real-proj: ℙ^n,
rneq: x ≠ y,
guard: {T},
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
req-vec: req-vec(n;x;y),
punit: u(a),
real-vec-mul: a*X,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
le: A ≤ B,
real-vec: ℝ^n,
rev_uimplies: rev_uimplies(P;Q),
less_than': less_than'(a;b),
true: True,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
rtermMinus: rtermMinus(num),
pi1: fst(t),
pi2: snd(t),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+
Lemmas referenced :
proj-sep_wf,
istype-void,
proj-eq_wf,
real-proj_wf,
istype-nat,
proj-eq-iff,
not-proj-sep,
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,
rdiv_wf,
real-vec-norm_wf,
proj-norm-positive,
rless_wf,
int-to-real_wf,
rneq_wf,
rmul_preserves_rneq_iff2,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
rneq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
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,
int_seg_wf,
rmul_preserves_req,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
req_functionality,
rminus_wf,
itermMinus_wf,
rminus_functionality,
real_term_value_minus_lemma,
rmul_reverses_rless_iff,
rless-int,
rless_functionality,
assert-rat-term-eq2,
rtermMinus_wf,
rtermMultiply_wf,
rtermVar_wf,
rtermDivide_wf,
rabs_wf,
real-vec-norm-mul,
real-vec-norm_functionality,
nat_plus_properties,
punit_wf,
rmul-rinv3,
rleq_weakening_rless,
rabs-of-nonpos,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
voidElimination,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
productElimination,
unionElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
inrFormation_alt,
isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
imageElimination,
applyEquality,
inlFormation_alt,
minusEquality,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbP{}\^{}n. (\mneg{}a \mneq{} b \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2020_05_20-PM-01_16_52
Last ObjectModification:
2019_12_10-AM-00_24_29
Theory : inner!product!spaces
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